Pre-Algebra Examples

$\begin{array}{c} x & y \\ - 2 & 20 \\ - 1 & 14 \\ 0 & 9.8 \\ 1 & 6.86 \\ 2 & 4.802 \end{array}$

Step 1

Check if the function rule is linear.

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Step 1.1

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Step 1.2

Build a set of equations from the table such that $y = a x + b$ .

$20 = a (- 2) + b 14 = a (- 1) + b 9.8 = a (0) + b 6.86 = a (1) + b 4.802 = a (2) + b$

Step 1.3

Calculate the values of $a$ and $b$ .

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Step 1.3.1

Rewrite the equation as $b = 9.8$ .

$b = 9.8$

$20 = a (- 2) + b$

$14 = a (- 1) + b$

$6.86 = a + b$

$4.802 = a (2) + b$

Step 1.3.2

Replace all occurrences of $b$ with $9.8$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $20 = a (- 2) + b$ with $9.8$ .

$20 = a (- 2) + 9.8$

$b = 9.8$

$14 = a (- 1) + b$

$6.86 = a + b$

$4.802 = a (2) + b$

Step 1.3.2.2

Simplify $20 = a (- 2) + 9.8$ .

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Step 1.3.2.2.1

Simplify the left side.

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Step 1.3.2.2.1.1

Remove parentheses.

$20 = a (- 2) + 9.8$

$b = 9.8$

$14 = a (- 1) + b$

$6.86 = a + b$

$4.802 = a (2) + b$

$20 = a (- 2) + 9.8$

$b = 9.8$

$14 = a (- 1) + b$

$6.86 = a + b$

$4.802 = a (2) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Move $- 2$ to the left of $a$ .

$20 = - 2 a + 9.8$

$b = 9.8$

$14 = a (- 1) + b$

$6.86 = a + b$

$4.802 = a (2) + b$

$20 = - 2 a + 9.8$

$b = 9.8$

$14 = a (- 1) + b$

$6.86 = a + b$

$4.802 = a (2) + b$

$20 = - 2 a + 9.8$

$b = 9.8$

$14 = a (- 1) + b$

$6.86 = a + b$

$4.802 = a (2) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $14 = a (- 1) + b$ with $9.8$ .

$14 = a (- 1) + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$6.86 = a + b$

$4.802 = a (2) + b$

Step 1.3.2.4

Simplify $14 = a (- 1) + 9.8$ .

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Step 1.3.2.4.1

Simplify the left side.

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Step 1.3.2.4.1.1

Remove parentheses.

$14 = a (- 1) + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$6.86 = a + b$

$4.802 = a (2) + b$

$14 = a (- 1) + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$6.86 = a + b$

$4.802 = a (2) + b$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify each term.

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Step 1.3.2.4.2.1.1

Move $- 1$ to the left of $a$ .

$14 = - 1 a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$6.86 = a + b$

$4.802 = a (2) + b$

Step 1.3.2.4.2.1.2

Rewrite $- 1 a$ as $- a$ .

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$6.86 = a + b$

$4.802 = a (2) + b$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$6.86 = a + b$

$4.802 = a (2) + b$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$6.86 = a + b$

$4.802 = a (2) + b$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$6.86 = a + b$

$4.802 = a (2) + b$

Step 1.3.2.5

Replace all occurrences of $b$ in $6.86 = a + b$ with $9.8$ .

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$4.802 = a (2) + b$

Step 1.3.2.6

Simplify the left side.

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Step 1.3.2.6.1

Remove parentheses.

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$4.802 = a (2) + b$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$4.802 = a (2) + b$

Step 1.3.2.7

Replace all occurrences of $b$ in $4.802 = a (2) + b$ with $9.8$ .

$4.802 = a (2) + 9.8$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.2.8

Simplify $4.802 = a (2) + 9.8$ .

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Step 1.3.2.8.1

Simplify the left side.

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Step 1.3.2.8.1.1

Remove parentheses.

$4.802 = a (2) + 9.8$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$4.802 = a (2) + 9.8$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.2.8.2

Simplify the right side.

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Step 1.3.2.8.2.1

Move $2$ to the left of $a$ .

$4.802 = 2 a + 9.8$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$4.802 = 2 a + 9.8$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$4.802 = 2 a + 9.8$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$4.802 = 2 a + 9.8$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.3

Solve for $a$ in $4.802 = 2 a + 9.8$ .

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Step 1.3.3.1

Rewrite the equation as $2 a + 9.8 = 4.802$ .

$2 a + 9.8 = 4.802$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.3.2

Move all terms not containing $a$ to the right side of the equation.

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Step 1.3.3.2.1

Subtract $9.8$ from both sides of the equation.

$2 a = 4.802 - 9.8$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.3.2.2

Subtract $9.8$ from $4.802$ .

$2 a = - 4.998$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$2 a = - 4.998$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.3.3

Divide each term in $2 a = - 4.998$ by $2$ and simplify.

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Step 1.3.3.3.1

Divide each term in $2 a = - 4.998$ by $2$ .

$\frac{2 a}{2} = \frac{- 4.998}{2}$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $2$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 4.998}{2}$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 4.998}{2}$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$a = \frac{- 4.998}{2}$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$a = \frac{- 4.998}{2}$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $- 4.998$ by $2$ .

$a = - 2.499$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$a = - 2.499$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$a = - 2.499$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$a = - 2.499$

$6.86 = a + 9.8$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.4

Replace all occurrences of $a$ with $- 2.499$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $6.86 = a + 9.8$ with $- 2.499$ .

$6.86 = (- 2.499) + 9.8$

$a = - 2.499$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Add $- 2.499$ and $9.8$ .

$6.86 = 7.301$

$a = - 2.499$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

$6.86 = 7.301$

$a = - 2.499$

$14 = - a + 9.8$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.4.3

Replace all occurrences of $a$ in $14 = - a + 9.8$ with $- 2.499$ .

$14 = - (- 2.499) + 9.8$

$6.86 = 7.301$

$a = - 2.499$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $- (- 2.499) + 9.8$ .

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Step 1.3.4.4.1.1

Multiply $- 1$ by $- 2.499$ .

$14 = 2.499 + 9.8$

$6.86 = 7.301$

$a = - 2.499$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.4.4.1.2

Add $2.499$ and $9.8$ .

$14 = 12.299$

$6.86 = 7.301$

$a = - 2.499$

$20 = - 2 a + 9.8$

$b = 9.8$

$14 = 12.299$

$6.86 = 7.301$

$a = - 2.499$

$20 = - 2 a + 9.8$

$b = 9.8$

$14 = 12.299$

$6.86 = 7.301$

$a = - 2.499$

$20 = - 2 a + 9.8$

$b = 9.8$

Step 1.3.4.5

Replace all occurrences of $a$ in $20 = - 2 a + 9.8$ with $- 2.499$ .

$20 = - 2 \cdot - 2.499 + 9.8$

$14 = 12.299$

$6.86 = 7.301$

$a = - 2.499$

$b = 9.8$

Step 1.3.4.6

Simplify the right side.

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Step 1.3.4.6.1

Simplify $- 2 \cdot - 2.499 + 9.8$ .

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Step 1.3.4.6.1.1

Multiply $- 2$ by $- 2.499$ .

$20 = 4.998 + 9.8$

$14 = 12.299$

$6.86 = 7.301$

$a = - 2.499$

$b = 9.8$

Step 1.3.4.6.1.2

Add $4.998$ and $9.8$ .

$20 = 14.798$

$14 = 12.299$

$6.86 = 7.301$

$a = - 2.499$

$b = 9.8$

$20 = 14.798$

$14 = 12.299$

$6.86 = 7.301$

$a = - 2.499$

$b = 9.8$

$20 = 14.798$

$14 = 12.299$

$6.86 = 7.301$

$a = - 2.499$

$b = 9.8$

$20 = 14.798$

$14 = 12.299$

$6.86 = 7.301$

$a = - 2.499$

$b = 9.8$

Step 1.3.5

Since $20 = 14.798$ is not true, there is no solution.

No solution

Step 1.4

Since $y \neq y$ for the corresponding $x$ values, the function is not linear.

The function is not linear

Step 2

Check if the function rule is quadratic.

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Step 2.1

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{2} + b x + c$ .

$y = a x^{2} + b x + c$

Step 2.2

Build a set of $3$ equations from the table such that $y = a x^{2} + b x + c$ .

Step 2.3

Calculate the values of $a$ , $b$ , and $c$ .

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Step 2.3.1

Solve for $c$ in $9.8 = a \cdot 0^{2} + c$ .

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Step 2.3.1.1

Rewrite the equation as $a \cdot 0^{2} + c = 9.8$ .

$a \cdot 0^{2} + c = 9.8$

$20 = a {(- 2)}^{2} + b (- 2) + c$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.1.2

Simplify $a \cdot 0^{2} + c$ .

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Step 2.3.1.2.1

Simplify each term.

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Step 2.3.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$a \cdot 0 + c = 9.8$

$20 = a {(- 2)}^{2} + b (- 2) + c$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.1.2.1.2

Multiply $a$ by $0$ .

$0 + c = 9.8$

$20 = a {(- 2)}^{2} + b (- 2) + c$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$0 + c = 9.8$

$20 = a {(- 2)}^{2} + b (- 2) + c$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.1.2.2

Add $0$ and $c$ .

$c = 9.8$

$20 = a {(- 2)}^{2} + b (- 2) + c$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$c = 9.8$

$20 = a {(- 2)}^{2} + b (- 2) + c$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$c = 9.8$

$20 = a {(- 2)}^{2} + b (- 2) + c$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2

Replace all occurrences of $c$ with $9.8$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $20 = a {(- 2)}^{2} + b (- 2) + c$ with $9.8$ .

$20 = a {(- 2)}^{2} + b (- 2) + 9.8$

$c = 9.8$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.2

Simplify $20 = a {(- 2)}^{2} + b (- 2) + 9.8$ .

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Step 2.3.2.2.1

Simplify the left side.

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Step 2.3.2.2.1.1

Remove parentheses.

$20 = a {(- 2)}^{2} + b (- 2) + 9.8$

$c = 9.8$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$20 = a {(- 2)}^{2} + b (- 2) + 9.8$

$c = 9.8$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify each term.

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Step 2.3.2.2.2.1.1

Raise $- 2$ to the power of $2$ .

$20 = a \cdot 4 + b (- 2) + 9.8$

$c = 9.8$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.2.2.1.2

Move $4$ to the left of $a$ .

$20 = 4 \cdot a + b (- 2) + 9.8$

$c = 9.8$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.2.2.1.3

Move $- 2$ to the left of $b$ .

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$14 = a {(- 1)}^{2} + b (- 1) + c$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $14 = a {(- 1)}^{2} + b (- 1) + c$ with $9.8$ .

$14 = a {(- 1)}^{2} + b (- 1) + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.4

Simplify $14 = a {(- 1)}^{2} + b (- 1) + 9.8$ .

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Step 2.3.2.4.1

Simplify the left side.

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Step 2.3.2.4.1.1

Remove parentheses.

$14 = a {(- 1)}^{2} + b (- 1) + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$14 = a {(- 1)}^{2} + b (- 1) + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify each term.

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Step 2.3.2.4.2.1.1

Raise $- 1$ to the power of $2$ .

$14 = a \cdot 1 + b (- 1) + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.4.2.1.2

Multiply $a$ by $1$ .

$14 = a + b (- 1) + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.4.2.1.3

Move $- 1$ to the left of $b$ .

$14 = a - 1 b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.4.2.1.4

Rewrite $- 1 b$ as $- b$ .

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$6.86 = a + b + c$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.5

Replace all occurrences of $c$ in $6.86 = a + b + c$ with $9.8$ .

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.6

Simplify the left side.

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Step 2.3.2.6.1

Remove parentheses.

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = a \cdot 2^{2} + b (2) + c$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = a \cdot 2^{2} + b (2) + c$

Step 2.3.2.7

Replace all occurrences of $c$ in $4.802 = a \cdot 2^{2} + b (2) + c$ with $9.8$ .

$4.802 = a \cdot 2^{2} + b (2) + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.2.8

Simplify $4.802 = a \cdot 2^{2} + b (2) + 9.8$ .

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Step 2.3.2.8.1

Simplify the left side.

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Step 2.3.2.8.1.1

Remove parentheses.

$4.802 = a \cdot 2^{2} + b (2) + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = a \cdot 2^{2} + b (2) + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.2.8.2

Simplify the right side.

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Step 2.3.2.8.2.1

Simplify each term.

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Step 2.3.2.8.2.1.1

Raise $2$ to the power of $2$ .

$4.802 = a \cdot 4 + b (2) + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.2.8.2.1.2

Move $4$ to the left of $a$ .

$4.802 = 4 \cdot a + b (2) + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.2.8.2.1.3

Move $2$ to the left of $b$ .

$4.802 = 4 a + 2 b + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = 4 a + 2 b + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = 4 a + 2 b + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = 4 a + 2 b + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = 4 a + 2 b + 9.8$

$6.86 = a + b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.3

Solve for $a$ in $6.86 = a + b + 9.8$ .

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Step 2.3.3.1

Rewrite the equation as $a + b + 9.8 = 6.86$ .

$a + b + 9.8 = 6.86$

$4.802 = 4 a + 2 b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.3.2

Move all terms not containing $a$ to the right side of the equation.

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Step 2.3.3.2.1

Subtract $b$ from both sides of the equation.

$a + 9.8 = 6.86 - b$

$4.802 = 4 a + 2 b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.3.2.2

Subtract $9.8$ from both sides of the equation.

$a = 6.86 - b - 9.8$

$4.802 = 4 a + 2 b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.3.2.3

Subtract $9.8$ from $6.86$ .

$a = - b - 2.94$

$4.802 = 4 a + 2 b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$a = - b - 2.94$

$4.802 = 4 a + 2 b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$a = - b - 2.94$

$4.802 = 4 a + 2 b + 9.8$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4

Replace all occurrences of $a$ with $- b - 2.94$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $4.802 = 4 a + 2 b + 9.8$ with $- b - 2.94$ .

$4.802 = 4 (- b - 2.94) + 2 b + 9.8$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $4 (- b - 2.94) + 2 b + 9.8$ .

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Step 2.3.4.2.1.1

Simplify each term.

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Step 2.3.4.2.1.1.1

Apply the distributive property.

$4.802 = 4 (- b) + 4 \cdot - 2.94 + 2 b + 9.8$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4.2.1.1.2

Multiply $- 1$ by $4$ .

$4.802 = - 4 b + 4 \cdot - 2.94 + 2 b + 9.8$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4.2.1.1.3

Multiply $4$ by $- 2.94$ .

$4.802 = - 4 b - 11.76 + 2 b + 9.8$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = - 4 b - 11.76 + 2 b + 9.8$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4.2.1.2

Simplify by adding terms.

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Step 2.3.4.2.1.2.1

Add $- 4 b$ and $2 b$ .

$4.802 = - 2 b - 11.76 + 9.8$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4.2.1.2.2

Add $- 11.76$ and $9.8$ .

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$14 = a - b + 9.8$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4.3

Replace all occurrences of $a$ in $14 = a - b + 9.8$ with $- b - 2.94$ .

$14 = (- b - 2.94) - b + 9.8$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $(- b - 2.94) - b + 9.8$ .

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Step 2.3.4.4.1.1

Subtract $b$ from $- b$ .

$14 = - 2 b - 2.94 + 9.8$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4.4.1.2

Add $- 2.94$ and $9.8$ .

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$20 = 4 a - 2 b + 9.8$

$c = 9.8$

Step 2.3.4.5

Replace all occurrences of $a$ in $20 = 4 a - 2 b + 9.8$ with $- b - 2.94$ .

$20 = 4 (- b - 2.94) - 2 b + 9.8$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.4.6

Simplify the right side.

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Step 2.3.4.6.1

Simplify $4 (- b - 2.94) - 2 b + 9.8$ .

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Step 2.3.4.6.1.1

Simplify each term.

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Step 2.3.4.6.1.1.1

Apply the distributive property.

$20 = 4 (- b) + 4 \cdot - 2.94 - 2 b + 9.8$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.4.6.1.1.2

Multiply $- 1$ by $4$ .

$20 = - 4 b + 4 \cdot - 2.94 - 2 b + 9.8$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.4.6.1.1.3

Multiply $4$ by $- 2.94$ .

$20 = - 4 b - 11.76 - 2 b + 9.8$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$20 = - 4 b - 11.76 - 2 b + 9.8$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.4.6.1.2

Simplify by adding terms.

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Step 2.3.4.6.1.2.1

Subtract $2 b$ from $- 4 b$ .

$20 = - 6 b - 11.76 + 9.8$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.4.6.1.2.2

Add $- 11.76$ and $9.8$ .

$20 = - 6 b - 1.96$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$20 = - 6 b - 1.96$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$20 = - 6 b - 1.96$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$20 = - 6 b - 1.96$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$20 = - 6 b - 1.96$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.5

Solve for $b$ in $20 = - 6 b - 1.96$ .

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Step 2.3.5.1

Rewrite the equation as $- 6 b - 1.96 = 20$ .

$- 6 b - 1.96 = 20$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.5.2

Move all terms not containing $b$ to the right side of the equation.

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Step 2.3.5.2.1

Add $1.96$ to both sides of the equation.

$- 6 b = 20 + 1.96$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.5.2.2

Add $20$ and $1.96$ .

$- 6 b = 21.96$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$- 6 b = 21.96$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.5.3

Divide each term in $- 6 b = 21.96$ by $- 6$ and simplify.

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Step 2.3.5.3.1

Divide each term in $- 6 b = 21.96$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{21.96}{- 6}$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.5.3.2

Simplify the left side.

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Step 2.3.5.3.2.1

Cancel the common factor of $- 6$ .

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Step 2.3.5.3.2.1.1

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{21.96}{- 6}$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.5.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{21.96}{- 6}$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$b = \frac{21.96}{- 6}$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$b = \frac{21.96}{- 6}$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide $21.96$ by $- 6$ .

$b = - 3.66$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$b = - 3.66$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$b = - 3.66$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$b = - 3.66$

$14 = - 2 b + 6.86$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.6

Replace all occurrences of $b$ with $- 3.66$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $14 = - 2 b + 6.86$ with $- 3.66$ .

$14 = - 2 \cdot - 3.66 + 6.86$

$b = - 3.66$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- 2 \cdot - 3.66 + 6.86$ .

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Step 2.3.6.2.1.1

Multiply $- 2$ by $- 3.66$ .

$14 = 7.32 + 6.86$

$b = - 3.66$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.6.2.1.2

Add $7.32$ and $6.86$ .

$14 = 14.18$

$b = - 3.66$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$14 = 14.18$

$b = - 3.66$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

$14 = 14.18$

$b = - 3.66$

$4.802 = - 2 b - 1.96$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.6.3

Replace all occurrences of $b$ in $4.802 = - 2 b - 1.96$ with $- 3.66$ .

$4.802 = - 2 \cdot - 3.66 - 1.96$

$14 = 14.18$

$b = - 3.66$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- 2 \cdot - 3.66 - 1.96$ .

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Step 2.3.6.4.1.1

Multiply $- 2$ by $- 3.66$ .

$4.802 = 7.32 - 1.96$

$14 = 14.18$

$b = - 3.66$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.6.4.1.2

Subtract $1.96$ from $7.32$ .

$4.802 = 5.36$

$14 = 14.18$

$b = - 3.66$

$a = - b - 2.94$

$c = 9.8$

$4.802 = 5.36$

$14 = 14.18$

$b = - 3.66$

$a = - b - 2.94$

$c = 9.8$

$4.802 = 5.36$

$14 = 14.18$

$b = - 3.66$

$a = - b - 2.94$

$c = 9.8$

Step 2.3.6.5

Replace all occurrences of $b$ in $a = - b - 2.94$ with $- 3.66$ .

$a = - (- 3.66) - 2.94$

$4.802 = 5.36$

$14 = 14.18$

$b = - 3.66$

$c = 9.8$

Step 2.3.6.6

Simplify the right side.

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Step 2.3.6.6.1

Simplify $- (- 3.66) - 2.94$ .

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Step 2.3.6.6.1.1

Multiply $- 1$ by $- 3.66$ .

$a = 3.66 - 2.94$

$4.802 = 5.36$

$14 = 14.18$

$b = - 3.66$

$c = 9.8$

Step 2.3.6.6.1.2

Subtract $2.94$ from $3.66$ .

$a = 0.72$

$4.802 = 5.36$

$14 = 14.18$

$b = - 3.66$

$c = 9.8$

$a = 0.72$

$4.802 = 5.36$

$14 = 14.18$

$b = - 3.66$

$c = 9.8$

$a = 0.72$

$4.802 = 5.36$

$14 = 14.18$

$b = - 3.66$

$c = 9.8$

$a = 0.72$

$4.802 = 5.36$

$14 = 14.18$

$b = - 3.66$

$c = 9.8$

Step 2.3.7

Since $4.802 = 5.36$ is not true, there is no solution.

No solution

Step 2.4

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $y$ value in the table.

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Step 2.4.1

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 0$ , $b = 0$ , $c = 0$ , and $x = - 2$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Raise $- 2$ to the power of $2$ .

$y = 0 \cdot 4 + (0) \cdot (- 2) + 0$

Step 2.4.1.1.2

Multiply $0$ by $4$ .

$y = 0 + (0) \cdot (- 2) + 0$

Step 2.4.1.1.3

Multiply $0$ by $- 2$ .

$y = 0 + 0 + 0$

Step 2.4.1.2

Simplify by adding numbers.

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Step 2.4.1.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 2.4.1.2.2

Add $0$ and $0$ .

$y = 0$

Step 2.4.2

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 2$ . This check does not pass, since $y = 0$ and $y = 20$ . The function rule can't be quadratic.

$0 \neq 20$

Step 2.4.3

Since $y \neq y$ for the corresponding $x$ values, the function is not quadratic.

The function is not quadratic

Step 3

Check if the function rule is cubic.

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Step 3.1

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{3} + b x^{2} + c x + d$ .

$y = a x^{3} + b x^{2} + c x + d$

Step 3.2

Build a set of $4$ equations from the table such that $y = a x^{3} + b x^{2} + c x + d$ .

Step 3.3

Calculate the values of $a$ , $b$ , $c$ , and $d$ .

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Step 3.3.1

Solve for $d$ in $20 = a {(- 2)}^{3} + b {(- 2)}^{2} + c (- 2) + d$ .

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Step 3.3.1.1

Rewrite the equation as $a {(- 2)}^{3} + b {(- 2)}^{2} + c (- 2) + d = 20$ .

$a {(- 2)}^{3} + b {(- 2)}^{2} + c (- 2) + d = 20$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.1.2

Simplify each term.

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Step 3.3.1.2.1

Raise $- 2$ to the power of $3$ .

$a \cdot - 8 + b {(- 2)}^{2} + c (- 2) + d = 20$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.1.2.2

Move $- 8$ to the left of $a$ .

$- 8 \cdot a + b {(- 2)}^{2} + c (- 2) + d = 20$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.1.2.3

Raise $- 2$ to the power of $2$ .

$- 8 a + b \cdot 4 + c (- 2) + d = 20$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.1.2.4

Move $4$ to the left of $b$ .

$- 8 a + 4 \cdot b + c (- 2) + d = 20$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.1.2.5

Move $- 2$ to the left of $c$ .

$- 8 a + 4 b - 2 c + d = 20$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$- 8 a + 4 b - 2 c + d = 20$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.1.3

Move all terms not containing $d$ to the right side of the equation.

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Step 3.3.1.3.1

Add $8 a$ to both sides of the equation.

$4 b - 2 c + d = 20 + 8 a$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.1.3.2

Subtract $4 b$ from both sides of the equation.

$- 2 c + d = 20 + 8 a - 4 b$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.1.3.3

Add $2 c$ to both sides of the equation.

$d = 20 + 8 a - 4 b + 2 c$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$d = 20 + 8 a - 4 b + 2 c$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$d = 20 + 8 a - 4 b + 2 c$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2

Replace all occurrences of $d$ with $20 + 8 a - 4 b + 2 c$ in each equation.

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Step 3.3.2.1

Replace all occurrences of $d$ in $14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + d$ with $20 + 8 a - 4 b + 2 c$ .

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2

Simplify $14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.2.1

Simplify the left side.

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Step 3.3.2.2.1.1

Remove parentheses.

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2

Simplify the right side.

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Step 3.3.2.2.2.1

Simplify $a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.2.2.1.1

Combine the opposite terms in $a {(- 1)}^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.2.2.1.1.1

Reorder the factors in the terms $c (- 2)$ and $2 c$ .

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} - 2 c + 20 + 8 a - 4 b + 2 c$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.1.2

Add $- 2 c$ and $2 c$ .

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b + 0$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.1.3

Add $a {(- 1)}^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b$ and $0$ .

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$14 = a {(- 1)}^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.2

Simplify each term.

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Step 3.3.2.2.2.1.2.1

Raise $- 1$ to the power of $3$ .

$14 = a \cdot - 1 + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.2.2

Move $- 1$ to the left of $a$ .

$14 = - 1 \cdot a + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.2.3

Rewrite $- 1 a$ as $- a$ .

$14 = - a + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.2.4

Raise $- 2$ to the power of $2$ .

$14 = - a + b \cdot 4 + 20 + 8 a - 4 b$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.2.5

Move $4$ to the left of $b$ .

$14 = - a + 4 b + 20 + 8 a - 4 b$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$14 = - a + 4 b + 20 + 8 a - 4 b$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.3

Simplify by adding terms.

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Step 3.3.2.2.2.1.3.1

Combine the opposite terms in $- a + 4 b + 20 + 8 a - 4 b$ .

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Step 3.3.2.2.2.1.3.1.1

Subtract $4 b$ from $4 b$ .

$14 = - a + 20 + 8 a + 0$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.3.1.2

Add $- a + 20 + 8 a$ and $0$ .

$14 = - a + 20 + 8 a$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$14 = - a + 20 + 8 a$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.2.2.1.3.2

Add $- a$ and $8 a$ .

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.3

Replace all occurrences of $d$ in $9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + d$ with $20 + 8 a - 4 b + 2 c$ .

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4

Simplify $9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.4.1

Simplify the left side.

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Step 3.3.2.4.1.1

Remove parentheses.

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2

Simplify the right side.

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Step 3.3.2.4.2.1

Simplify $a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.4.2.1.1

Combine the opposite terms in $a \cdot 0^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.4.2.1.1.1

Reorder the factors in the terms $c (- 2)$ and $2 c$ .

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} - 2 c + 20 + 8 a - 4 b + 2 c$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2.1.1.2

Add $- 2 c$ and $2 c$ .

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b + 0$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2.1.1.3

Add $a \cdot 0^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b$ and $0$ .

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = a \cdot 0^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2.1.2

Simplify each term.

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Step 3.3.2.4.2.1.2.1

Raising $0$ to any positive power yields $0$ .

$9.8 = a \cdot 0 + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2.1.2.2

Multiply $a$ by $0$ .

$9.8 = 0 + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2.1.2.3

Raise $- 2$ to the power of $2$ .

$9.8 = 0 + b \cdot 4 + 20 + 8 a - 4 b$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2.1.2.4

Move $4$ to the left of $b$ .

$9.8 = 0 + 4 b + 20 + 8 a - 4 b$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = 0 + 4 b + 20 + 8 a - 4 b$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2.1.3

Combine the opposite terms in $0 + 4 b + 20 + 8 a - 4 b$ .

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Step 3.3.2.4.2.1.3.1

Add $0$ and $4 b$ .

$9.8 = 4 b + 20 + 8 a - 4 b$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2.1.3.2

Subtract $4 b$ from $4 b$ .

$9.8 = 20 + 8 a + 0$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.4.2.1.3.3

Add $20 + 8 a$ and $0$ .

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + d$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.5

Replace all occurrences of $d$ in $6.86 = a + b {(- 2)}^{2} + c (- 2) + d$ with $20 + 8 a - 4 b + 2 c$ .

$6.86 = a + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.6

Simplify $6.86 = a + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.6.1

Simplify the left side.

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Step 3.3.2.6.1.1

Remove parentheses.

$6.86 = a + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.6.2

Simplify the right side.

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Step 3.3.2.6.2.1

Simplify $a + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.6.2.1.1

Combine the opposite terms in $a + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.6.2.1.1.1

Reorder the factors in the terms $c (- 2)$ and $2 c$ .

$6.86 = a + b {(- 2)}^{2} - 2 c + 20 + 8 a - 4 b + 2 c$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.6.2.1.1.2

Add $- 2 c$ and $2 c$ .

$6.86 = a + b {(- 2)}^{2} + 20 + 8 a - 4 b + 0$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.6.2.1.1.3

Add $a + b {(- 2)}^{2} + 20 + 8 a - 4 b$ and $0$ .

$6.86 = a + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.6.2.1.2

Simplify each term.

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Step 3.3.2.6.2.1.2.1

Raise $- 2$ to the power of $2$ .

$6.86 = a + b \cdot 4 + 20 + 8 a - 4 b$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.6.2.1.2.2

Move $4$ to the left of $b$ .

$6.86 = a + 4 b + 20 + 8 a - 4 b$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + 4 b + 20 + 8 a - 4 b$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.6.2.1.3

Simplify by adding terms.

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Step 3.3.2.6.2.1.3.1

Combine the opposite terms in $a + 4 b + 20 + 8 a - 4 b$ .

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Step 3.3.2.6.2.1.3.1.1

Subtract $4 b$ from $4 b$ .

$6.86 = a + 20 + 8 a + 0$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.6.2.1.3.1.2

Add $a + 20 + 8 a$ and $0$ .

$6.86 = a + 20 + 8 a$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = a + 20 + 8 a$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.6.2.1.3.2

Add $a$ and $8 a$ .

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$

Step 3.3.2.7

Replace all occurrences of $d$ in $4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + d$ with $20 + 8 a - 4 b + 2 c$ .

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8

Simplify $4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.8.1

Simplify the left side.

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Step 3.3.2.8.1.1

Remove parentheses.

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2

Simplify the right side.

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Step 3.3.2.8.2.1

Simplify $a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.8.2.1.1

Combine the opposite terms in $a \cdot 2^{3} + b {(- 2)}^{2} + c (- 2) + 20 + 8 a - 4 b + 2 c$ .

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Step 3.3.2.8.2.1.1.1

Reorder the factors in the terms $c (- 2)$ and $2 c$ .

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} - 2 c + 20 + 8 a - 4 b + 2 c$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2.1.1.2

Add $- 2 c$ and $2 c$ .

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b + 0$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2.1.1.3

Add $a \cdot 2^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b$ and $0$ .

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = a \cdot 2^{3} + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2.1.2

Simplify each term.

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Step 3.3.2.8.2.1.2.1

Raise $2$ to the power of $3$ .

$4.802 = a \cdot 8 + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2.1.2.2

Move $8$ to the left of $a$ .

$4.802 = 8 \cdot a + b {(- 2)}^{2} + 20 + 8 a - 4 b$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2.1.2.3

Raise $- 2$ to the power of $2$ .

$4.802 = 8 a + b \cdot 4 + 20 + 8 a - 4 b$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2.1.2.4

Move $4$ to the left of $b$ .

$4.802 = 8 a + 4 b + 20 + 8 a - 4 b$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = 8 a + 4 b + 20 + 8 a - 4 b$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2.1.3

Simplify by adding terms.

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Step 3.3.2.8.2.1.3.1

Combine the opposite terms in $8 a + 4 b + 20 + 8 a - 4 b$ .

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Step 3.3.2.8.2.1.3.1.1

Subtract $4 b$ from $4 b$ .

$4.802 = 8 a + 20 + 8 a + 0$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2.1.3.1.2

Add $8 a + 20 + 8 a$ and $0$ .

$4.802 = 8 a + 20 + 8 a$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = 8 a + 20 + 8 a$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.2.8.2.1.3.2

Add $8 a$ and $8 a$ .

$4.802 = 16 a + 20$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = 16 a + 20$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = 16 a + 20$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = 16 a + 20$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = 16 a + 20$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$4.802 = 16 a + 20$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.3

Solve for $a$ in $4.802 = 16 a + 20$ .

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Step 3.3.3.1

Rewrite the equation as $16 a + 20 = 4.802$ .

$16 a + 20 = 4.802$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.3.2

Move all terms not containing $a$ to the right side of the equation.

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Step 3.3.3.2.1

Subtract $20$ from both sides of the equation.

$16 a = 4.802 - 20$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.3.2.2

Subtract $20$ from $4.802$ .

$16 a = - 15.198$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$16 a = - 15.198$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.3.3

Divide each term in $16 a = - 15.198$ by $16$ and simplify.

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Step 3.3.3.3.1

Divide each term in $16 a = - 15.198$ by $16$ .

$\frac{16 a}{16} = \frac{- 15.198}{16}$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.3.3.2

Simplify the left side.

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Step 3.3.3.3.2.1

Cancel the common factor of $16$ .

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Step 3.3.3.3.2.1.1

Cancel the common factor.

$\frac{16 a}{16} = \frac{- 15.198}{16}$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 15.198}{16}$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$a = \frac{- 15.198}{16}$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$a = \frac{- 15.198}{16}$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.3.3.3

Simplify the right side.

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Step 3.3.3.3.3.1

Divide $- 15.198$ by $16$ .

$a = - 0.949875$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$a = - 0.949875$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$a = - 0.949875$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$a = - 0.949875$

$6.86 = 9 a + 20$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4

Replace all occurrences of $a$ with $- 0.949875$ in each equation.

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Step 3.3.4.1

Replace all occurrences of $a$ in $6.86 = 9 a + 20$ with $- 0.949875$ .

$6.86 = 9 (- 0.949875) + 20$

$a = - 0.949875$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4.2

Simplify the right side.

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Step 3.3.4.2.1

Simplify $9 (- 0.949875) + 20$ .

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Step 3.3.4.2.1.1

Multiply $9$ by $- 0.949875$ .

$6.86 = - 8.548875 + 20$

$a = - 0.949875$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4.2.1.2

Add $- 8.548875$ and $20$ .

$6.86 = 11.451125$

$a = - 0.949875$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = 11.451125$

$a = - 0.949875$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$6.86 = 11.451125$

$a = - 0.949875$

$9.8 = 20 + 8 a$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4.3

Replace all occurrences of $a$ in $9.8 = 20 + 8 a$ with $- 0.949875$ .

$9.8 = 20 + 8 (- 0.949875)$

$6.86 = 11.451125$

$a = - 0.949875$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4.4

Simplify the right side.

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Step 3.3.4.4.1

Simplify $20 + 8 (- 0.949875)$ .

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Step 3.3.4.4.1.1

Multiply $8$ by $- 0.949875$ .

$9.8 = 20 - 7.599$

$6.86 = 11.451125$

$a = - 0.949875$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4.4.1.2

Subtract $7.599$ from $20$ .

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$14 = 7 a + 20$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4.5

Replace all occurrences of $a$ in $14 = 7 a + 20$ with $- 0.949875$ .

$14 = 7 (- 0.949875) + 20$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4.6

Simplify the right side.

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Step 3.3.4.6.1

Simplify $7 (- 0.949875) + 20$ .

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Step 3.3.4.6.1.1

Multiply $7$ by $- 0.949875$ .

$14 = - 6.649125 + 20$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4.6.1.2

Add $- 6.649125$ and $20$ .

$14 = 13.350875$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$d = 20 + 8 a - 4 b + 2 c$

$14 = 13.350875$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$d = 20 + 8 a - 4 b + 2 c$

$14 = 13.350875$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$d = 20 + 8 a - 4 b + 2 c$

Step 3.3.4.7

Replace all occurrences of $a$ in $d = 20 + 8 a - 4 b + 2 c$ with $- 0.949875$ .

$d = 20 + 8 (- 0.949875) - 4 b + 2 c$

$14 = 13.350875$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

Step 3.3.4.8

Simplify the right side.

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Step 3.3.4.8.1

Simplify $20 + 8 (- 0.949875) - 4 b + 2 c$ .

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Step 3.3.4.8.1.1

Multiply $8$ by $- 0.949875$ .

$d = 20 - 7.599 - 4 b + 2 c$

$14 = 13.350875$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

Step 3.3.4.8.1.2

Subtract $7.599$ from $20$ .

$d = 12.401 - 4 b + 2 c$

$14 = 13.350875$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$d = 12.401 - 4 b + 2 c$

$14 = 13.350875$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$d = 12.401 - 4 b + 2 c$

$14 = 13.350875$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

$d = 12.401 - 4 b + 2 c$

$14 = 13.350875$

$9.8 = 12.401$

$6.86 = 11.451125$

$a = - 0.949875$

Step 3.3.5

Since $14 = 13.350875$ is not true, there is no solution.

No solution

Step 3.4

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $y$ value in the table.

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Step 3.4.1

Calculate the value of $y$ such that $y = a x^{3} + b$ when $a = 0$ , $b = 0$ , $c = 0$ , $d = 0$ , and $x = - 2$ .

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Step 3.4.1.1

Simplify each term.

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Step 3.4.1.1.1

Raise $- 2$ to the power of $3$ .

$y = 0 \cdot - 8 + (0) \cdot (- 2^{2}) + (0) \cdot ((- 2) \cdot 0)$

Step 3.4.1.1.2

Multiply $0$ by $- 8$ .

$y = 0 + (0) \cdot (- 2^{2}) + (0) \cdot ((- 2) \cdot 0)$

Step 3.4.1.1.3

Raise $2$ to the power of $2$ .

$y = 0 + 0 \cdot (- 1 \cdot 4) + (0) \cdot ((- 2) \cdot 0)$

Step 3.4.1.1.4

Multiply $0 (- 1 \cdot 4)$ .

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Step 3.4.1.1.4.1

Multiply $- 1$ by $4$ .

$y = 0 + 0 \cdot - 4 + (0) \cdot ((- 2) \cdot 0)$

Step 3.4.1.1.4.2

Multiply $0$ by $- 4$ .

$y = 0 + 0 + (0) \cdot ((- 2) \cdot 0)$

Step 3.4.1.1.5

Multiply $(0) ((- 2) \cdot 0)$ .

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Step 3.4.1.1.5.1

Multiply $- 2$ by $0$ .

$y = 0 + 0 + 0 \cdot 0$

Step 3.4.1.1.5.2

Multiply $0$ by $0$ .

$y = 0 + 0 + 0$

Step 3.4.1.2

Simplify by adding numbers.

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Step 3.4.1.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 3.4.1.2.2

Add $0$ and $0$ .

$y = 0$

Step 3.4.2

If the table has a cubic function rule, $y = y$ for the corresponding $x$ value, $x = - 2$ . This check does not pass, since $y = 0$ and $y = 20$ . The function rule can't be cubic.

$0 \neq 20$

Step 3.4.3

Since $y \neq y$ for the corresponding $x$ values, the function is not cubic.

The function is not cubic

Step 4

There are no values for $a$ , $b$ , $c$ , and $d$ in the equations $y = a x + b$ , $y = a x^{2} + b x + c$ , and $y = a x^{3} + b x^{2} + c x + d$ that work for every pair of $x$ and $y$ .

The table does not have a function rule that is linear, quadratic, or cubic.