Algebra Examples

$\begin{array}{c} x & f (x) \\ 3 & 64 \\ 4 & 256 \\ 5 & 1024 \end{array}$

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 $f (x) = a x + b$ .

$64 = a (3) + b 256 = a (4) + b 1024 = a (5) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $64 = a (3) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (3) + b = 64$ .

$a (3) + b = 64$

$256 = a (4) + b$

$1024 = a (5) + b$

Step 1.3.1.2

Move $3$ to the left of $a$ .

$3 a + b = 64$

$256 = a (4) + b$

$1024 = a (5) + b$

Step 1.3.1.3

Subtract $3 a$ from both sides of the equation.

$b = 64 - 3 a$

$256 = a (4) + b$

$1024 = a (5) + b$

$b = 64 - 3 a$

$256 = a (4) + b$

$1024 = a (5) + b$

Step 1.3.2

Replace all occurrences of $b$ with $64 - 3 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $256 = a (4) + b$ with $64 - 3 a$ .

$256 = a (4) + 64 - 3 a$

$b = 64 - 3 a$

$1024 = a (5) + b$

Step 1.3.2.2

Simplify $256 = a (4) + 64 - 3 a$ .

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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.

$256 = a (4) + 64 - 3 a$

$b = 64 - 3 a$

$1024 = a (5) + b$

$256 = a (4) + 64 - 3 a$

$b = 64 - 3 a$

$1024 = a (5) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (4) + 64 - 3 a$ .

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Step 1.3.2.2.2.1.1

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

$256 = 4 a + 64 - 3 a$

$b = 64 - 3 a$

$1024 = a (5) + b$

Step 1.3.2.2.2.1.2

Subtract $3 a$ from $4 a$ .

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + b$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + b$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + b$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $1024 = a (5) + b$ with $64 - 3 a$ .

$1024 = a (5) + 64 - 3 a$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.2.4

Simplify $1024 = a (5) + 64 - 3 a$ .

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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.

$1024 = a (5) + 64 - 3 a$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = a (5) + 64 - 3 a$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (5) + 64 - 3 a$ .

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Step 1.3.2.4.2.1.1

Move $5$ to the left of $a$ .

$1024 = 5 a + 64 - 3 a$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.2.4.2.1.2

Subtract $3 a$ from $5 a$ .

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

$1024 = 2 a + 64$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3

Solve for $a$ in $1024 = 2 a + 64$ .

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Step 1.3.3.1

Rewrite the equation as $2 a + 64 = 1024$ .

$2 a + 64 = 1024$

$256 = a + 64$

$b = 64 - 3 a$

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 $64$ from both sides of the equation.

$2 a = 1024 - 64$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.2.2

Subtract $64$ from $1024$ .

$2 a = 960$

$256 = a + 64$

$b = 64 - 3 a$

$2 a = 960$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.3

Divide each term in $2 a = 960$ by $2$ and simplify.

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Step 1.3.3.3.1

Divide each term in $2 a = 960$ by $2$ .

$\frac{2 a}{2} = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

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.

$\frac{2 a}{2} = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

$a = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

$a = \frac{960}{2}$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $960$ by $2$ .

$a = 480$

$256 = a + 64$

$b = 64 - 3 a$

$a = 480$

$256 = a + 64$

$b = 64 - 3 a$

$a = 480$

$256 = a + 64$

$b = 64 - 3 a$

$a = 480$

$256 = a + 64$

$b = 64 - 3 a$

Step 1.3.4

Replace all occurrences of $a$ with $480$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $256 = a + 64$ with $480$ .

$256 = (480) + 64$

$a = 480$

$b = 64 - 3 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Add $480$ and $64$ .

$256 = 544$

$a = 480$

$b = 64 - 3 a$

$256 = 544$

$a = 480$

$b = 64 - 3 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = 64 - 3 a$ with $480$ .

$b = 64 - 3 \cdot 480$

$256 = 544$

$a = 480$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $64 - 3 \cdot 480$ .

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Step 1.3.4.4.1.1

Multiply $- 3$ by $480$ .

$b = 64 - 1440$

$256 = 544$

$a = 480$

Step 1.3.4.4.1.2

Subtract $1440$ from $64$ .

$b = - 1376$

$256 = 544$

$a = 480$

$b = - 1376$

$256 = 544$

$a = 480$

$b = - 1376$

$256 = 544$

$a = 480$

$b = - 1376$

$256 = 544$

$a = 480$

Step 1.3.5

Since $256 = 544$ is not true, there is no solution.

No solution

Step 1.4

Since $y \neq f (x)$ 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 $f (x) = a x^{2} + b x + c$ .

Step 2.3

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

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Step 2.3.1

Solve for $c$ in $64 = a \cdot 3^{2} + b (3) + c$ .

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Step 2.3.1.1

Rewrite the equation as $a \cdot 3^{2} + b (3) + c = 64$ .

$a \cdot 3^{2} + b (3) + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

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

$a \cdot 9 + b (3) + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2.2

Move $9$ to the left of $a$ .

$9 \cdot a + b (3) + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2.3

Move $3$ to the left of $b$ .

$9 a + 3 b + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

$9 a + 3 b + c = 64$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.3

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

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Step 2.3.1.3.1

Subtract $9 a$ from both sides of the equation.

$3 b + c = 64 - 9 a$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.3.2

Subtract $3 b$ from both sides of the equation.

$c = 64 - 9 a - 3 b$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

$c = 64 - 9 a - 3 b$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

$c = 64 - 9 a - 3 b$

$256 = a \cdot 4^{2} + b (4) + c$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2

Replace all occurrences of $c$ with $64 - 9 a - 3 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $256 = a \cdot 4^{2} + b (4) + c$ with $64 - 9 a - 3 b$ .

$256 = a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2

Simplify $256 = a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$ .

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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.

$256 = a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify $a \cdot 4^{2} + b (4) + 64 - 9 a - 3 b$ .

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Step 2.3.2.2.2.1.1

Simplify each term.

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Step 2.3.2.2.2.1.1.1

Raise $4$ to the power of $2$ .

$256 = a \cdot 16 + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.1.2

Move $16$ to the left of $a$ .

$256 = 16 \cdot a + b (4) + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.1.3

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

$256 = 16 a + 4 b + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 16 a + 4 b + 64 - 9 a - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.2

Simplify by adding terms.

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Step 2.3.2.2.2.1.2.1

Subtract $9 a$ from $16 a$ .

$256 = 7 a + 4 b + 64 - 3 b$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.2.2

Subtract $3 b$ from $4 b$ .

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $1024 = a \cdot 5^{2} + b (5) + c$ with $64 - 9 a - 3 b$ .

$1024 = a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4

Simplify $1024 = a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$ .

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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.

$1024 = a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify $a \cdot 5^{2} + b (5) + 64 - 9 a - 3 b$ .

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Step 2.3.2.4.2.1.1

Simplify each term.

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Step 2.3.2.4.2.1.1.1

Raise $5$ to the power of $2$ .

$1024 = a \cdot 25 + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2.1.1.2

Move $25$ to the left of $a$ .

$1024 = 25 \cdot a + b (5) + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2.1.1.3

Move $5$ to the left of $b$ .

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 25 a + 5 b + 64 - 9 a - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2.1.2

Simplify by adding terms.

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Step 2.3.2.4.2.1.2.1

Subtract $9 a$ from $25 a$ .

$1024 = 16 a + 5 b + 64 - 3 b$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.2.4.2.1.2.2

Subtract $3 b$ from $5 b$ .

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

$1024 = 16 a + 2 b + 64$

$256 = 7 a + b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.3

Solve for $b$ in $256 = 7 a + b + 64$ .

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Step 2.3.3.1

Rewrite the equation as $7 a + b + 64 = 256$ .

$7 a + b + 64 = 256$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.3.2

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

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Step 2.3.3.2.1

Subtract $7 a$ from both sides of the equation.

$b + 64 = 256 - 7 a$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.3.2.2

Subtract $64$ from both sides of the equation.

$b = 256 - 7 a - 64$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.3.2.3

Subtract $64$ from $256$ .

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

$b = - 7 a + 192$

$1024 = 16 a + 2 b + 64$

$c = 64 - 9 a - 3 b$

Step 2.3.4

Replace all occurrences of $b$ with $- 7 a + 192$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $b$ in $1024 = 16 a + 2 b + 64$ with $- 7 a + 192$ .

$1024 = 16 a + 2 (- 7 a + 192) + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $16 a + 2 (- 7 a + 192) + 64$ .

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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.

$1024 = 16 a + 2 (- 7 a) + 2 \cdot 192 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2.1.1.2

Multiply $- 7$ by $2$ .

$1024 = 16 a - 14 a + 2 \cdot 192 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2.1.1.3

Multiply $2$ by $192$ .

$1024 = 16 a - 14 a + 384 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

$1024 = 16 a - 14 a + 384 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2.1.2

Simplify by adding terms.

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Step 2.3.4.2.1.2.1

Subtract $14 a$ from $16 a$ .

$1024 = 2 a + 384 + 64$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.2.1.2.2

Add $384$ and $64$ .

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a - 3 b$

Step 2.3.4.3

Replace all occurrences of $b$ in $c = 64 - 9 a - 3 b$ with $- 7 a + 192$ .

$c = 64 - 9 a - 3 (- 7 a + 192)$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $64 - 9 a - 3 (- 7 a + 192)$ .

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Step 2.3.4.4.1.1

Simplify each term.

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Step 2.3.4.4.1.1.1

Apply the distributive property.

$c = 64 - 9 a - 3 (- 7 a) - 3 \cdot 192$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4.1.1.2

Multiply $- 7$ by $- 3$ .

$c = 64 - 9 a + 21 a - 3 \cdot 192$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4.1.1.3

Multiply $- 3$ by $192$ .

$c = 64 - 9 a + 21 a - 576$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 64 - 9 a + 21 a - 576$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4.1.2

Simplify by adding terms.

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Step 2.3.4.4.1.2.1

Subtract $576$ from $64$ .

$c = - 9 a + 21 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.4.4.1.2.2

Add $- 9 a$ and $21 a$ .

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

$c = 12 a - 512$

$1024 = 2 a + 448$

$b = - 7 a + 192$

Step 2.3.5

Solve for $a$ in $1024 = 2 a + 448$ .

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Step 2.3.5.1

Rewrite the equation as $2 a + 448 = 1024$ .

$2 a + 448 = 1024$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.2

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

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Step 2.3.5.2.1

Subtract $448$ from both sides of the equation.

$2 a = 1024 - 448$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.2.2

Subtract $448$ from $1024$ .

$2 a = 576$

$c = 12 a - 512$

$b = - 7 a + 192$

$2 a = 576$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.3

Divide each term in $2 a = 576$ by $2$ and simplify.

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Step 2.3.5.3.1

Divide each term in $2 a = 576$ by $2$ .

$\frac{2 a}{2} = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

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 $2$ .

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Step 2.3.5.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = \frac{576}{2}$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide $576$ by $2$ .

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

$a = 288$

$c = 12 a - 512$

$b = - 7 a + 192$

Step 2.3.6

Replace all occurrences of $a$ with $288$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $a$ in $c = 12 a - 512$ with $288$ .

$c = 12 (288) - 512$

$a = 288$

$b = - 7 a + 192$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $12 (288) - 512$ .

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Step 2.3.6.2.1.1

Multiply $12$ by $288$ .

$c = 3456 - 512$

$a = 288$

$b = - 7 a + 192$

Step 2.3.6.2.1.2

Subtract $512$ from $3456$ .

$c = 2944$

$a = 288$

$b = - 7 a + 192$

$c = 2944$

$a = 288$

$b = - 7 a + 192$

$c = 2944$

$a = 288$

$b = - 7 a + 192$

Step 2.3.6.3

Replace all occurrences of $a$ in $b = - 7 a + 192$ with $288$ .

$b = - 7 \cdot 288 + 192$

$c = 2944$

$a = 288$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- 7 \cdot 288 + 192$ .

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Step 2.3.6.4.1.1

Multiply $- 7$ by $288$ .

$b = - 2016 + 192$

$c = 2944$

$a = 288$

Step 2.3.6.4.1.2

Add $- 2016$ and $192$ .

$b = - 1824$

$c = 2944$

$a = 288$

$b = - 1824$

$c = 2944$

$a = 288$

$b = - 1824$

$c = 2944$

$a = 288$

$b = - 1824$

$c = 2944$

$a = 288$

Step 2.3.7

List all of the solutions.

$b = - 1824, c = 2944, a = 288$

Step 2.4

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $f (x)$ 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 $a = 288$ , $b = - 1824$ , $c = 2944$ , and $x = 3$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

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

$y = 288 \cdot 9 + (- 1824) \cdot (3) + 2944$

Step 2.4.1.1.2

Multiply $288$ by $9$ .

$y = 2592 + (- 1824) \cdot (3) + 2944$

Step 2.4.1.1.3

Multiply $- 1824$ by $3$ .

$y = 2592 - 5472 + 2944$

Step 2.4.1.2

Simplify by adding and subtracting.

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Step 2.4.1.2.1

Subtract $5472$ from $2592$ .

$y = - 2880 + 2944$

Step 2.4.1.2.2

Add $- 2880$ and $2944$ .

$y = 64$

Step 2.4.2

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 64$ and $f (x) = 64$ .

$64 = 64$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 288$ , $b = - 1824$ , $c = 2944$ , and $x = 4$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Raise $4$ to the power of $2$ .

$y = 288 \cdot 16 + (- 1824) \cdot (4) + 2944$

Step 2.4.3.1.2

Multiply $288$ by $16$ .

$y = 4608 + (- 1824) \cdot (4) + 2944$

Step 2.4.3.1.3

Multiply $- 1824$ by $4$ .

$y = 4608 - 7296 + 2944$

Step 2.4.3.2

Simplify by adding and subtracting.

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Step 2.4.3.2.1

Subtract $7296$ from $4608$ .

$y = - 2688 + 2944$

Step 2.4.3.2.2

Add $- 2688$ and $2944$ .

$y = 256$

Step 2.4.4

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 256$ and $f (x) = 256$ .

$256 = 256$

Step 2.4.5

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 288$ , $b = - 1824$ , $c = 2944$ , and $x = 5$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Raise $5$ to the power of $2$ .

$y = 288 \cdot 25 + (- 1824) \cdot (5) + 2944$

Step 2.4.5.1.2

Multiply $288$ by $25$ .

$y = 7200 + (- 1824) \cdot (5) + 2944$

Step 2.4.5.1.3

Multiply $- 1824$ by $5$ .

$y = 7200 - 9120 + 2944$

Step 2.4.5.2

Simplify by adding and subtracting.

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Step 2.4.5.2.1

Subtract $9120$ from $7200$ .

$y = - 1920 + 2944$

Step 2.4.5.2.2

Add $- 1920$ and $2944$ .

$y = 1024$

Step 2.4.6

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 5$ . This check passes since $y = 1024$ and $f (x) = 1024$ .

$1024 = 1024$

Step 2.4.7

Since $y = f (x)$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Step 3

Since all $y = f (x)$ , the function is quadratic and follows the form $y = 288 x^{2} - 1824 x + 2944$ .

$y = 288 x^{2} - 1824 x + 2944$