Finite Math Examples

$2 x^{2} + 4 x - y = 6$ , $2 x - y = - 6$

Step 1

Solve for $y$ in $2 x^{2} + 4 x - y = 6$ .

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Step 1.1

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

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Step 1.1.1

Subtract $2 x^{2}$ from both sides of the equation.

$4 x - y = 6 - 2 x^{2}$

$2 x - y = - 6$

Step 1.1.2

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

$- y = 6 - 2 x^{2} - 4 x$

$2 x - y = - 6$

$- y = 6 - 2 x^{2} - 4 x$

$2 x - y = - 6$

Step 1.2

Divide each term in $- y = 6 - 2 x^{2} - 4 x$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- y = 6 - 2 x^{2} - 4 x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{6}{- 1} + \frac{- 2 x^{2}}{- 1} + \frac{- 4 x}{- 1}$

$2 x - y = - 6$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{6}{- 1} + \frac{- 2 x^{2}}{- 1} + \frac{- 4 x}{- 1}$

$2 x - y = - 6$

Step 1.2.2.2

Divide $y$ by $1$ .

$y = \frac{6}{- 1} + \frac{- 2 x^{2}}{- 1} + \frac{- 4 x}{- 1}$

$2 x - y = - 6$

$y = \frac{6}{- 1} + \frac{- 2 x^{2}}{- 1} + \frac{- 4 x}{- 1}$

$2 x - y = - 6$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $6$ by $- 1$ .

$y = - 6 + \frac{- 2 x^{2}}{- 1} + \frac{- 4 x}{- 1}$

$2 x - y = - 6$

Step 1.2.3.1.2

Move the negative one from the denominator of $\frac{- 2 x^{2}}{- 1}$ .

$y = - 6 - 1 \cdot (- 2 x^{2}) + \frac{- 4 x}{- 1}$

$2 x - y = - 6$

Step 1.2.3.1.3

Rewrite $- 1 \cdot (- 2 x^{2})$ as $- (- 2 x^{2})$ .

$y = - 6 - (- 2 x^{2}) + \frac{- 4 x}{- 1}$

$2 x - y = - 6$

Step 1.2.3.1.4

Multiply $- 2$ by $- 1$ .

$y = - 6 + 2 x^{2} + \frac{- 4 x}{- 1}$

$2 x - y = - 6$

Step 1.2.3.1.5

Move the negative one from the denominator of $\frac{- 4 x}{- 1}$ .

$y = - 6 + 2 x^{2} - 1 \cdot (- 4 x)$

$2 x - y = - 6$

Step 1.2.3.1.6

Rewrite $- 1 \cdot (- 4 x)$ as $- (- 4 x)$ .

$y = - 6 + 2 x^{2} - (- 4 x)$

$2 x - y = - 6$

Step 1.2.3.1.7

Multiply $- 4$ by $- 1$ .

$y = - 6 + 2 x^{2} + 4 x$

$2 x - y = - 6$

$y = - 6 + 2 x^{2} + 4 x$

$2 x - y = - 6$

$y = - 6 + 2 x^{2} + 4 x$

$2 x - y = - 6$

$y = - 6 + 2 x^{2} + 4 x$

$2 x - y = - 6$

$y = - 6 + 2 x^{2} + 4 x$

$2 x - y = - 6$

Step 2

Replace all occurrences of $y$ with $- 6 + 2 x^{2} + 4 x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x - y = - 6$ with $- 6 + 2 x^{2} + 4 x$ .

$2 x - (- 6 + 2 x^{2} + 4 x) = - 6$

$y = - 6 + 2 x^{2} + 4 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 x - (- 6 + 2 x^{2} + 4 x)$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$2 x + 6 - (2 x^{2}) - (4 x) = - 6$

$y = - 6 + 2 x^{2} + 4 x$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 6$ .

$2 x + 6 - (2 x^{2}) - (4 x) = - 6$

$y = - 6 + 2 x^{2} + 4 x$

Step 2.2.1.1.2.2

Multiply $2$ by $- 1$ .

$2 x + 6 - 2 x^{2} - (4 x) = - 6$

$y = - 6 + 2 x^{2} + 4 x$

Step 2.2.1.1.2.3

Multiply $4$ by $- 1$ .

$2 x + 6 - 2 x^{2} - 4 x = - 6$

$y = - 6 + 2 x^{2} + 4 x$

$2 x + 6 - 2 x^{2} - 4 x = - 6$

$y = - 6 + 2 x^{2} + 4 x$

$2 x + 6 - 2 x^{2} - 4 x = - 6$

$y = - 6 + 2 x^{2} + 4 x$

Step 2.2.1.2

Subtract $4 x$ from $2 x$ .

$- 2 x + 6 - 2 x^{2} = - 6$

$y = - 6 + 2 x^{2} + 4 x$

$- 2 x + 6 - 2 x^{2} = - 6$

$y = - 6 + 2 x^{2} + 4 x$

$- 2 x + 6 - 2 x^{2} = - 6$

$y = - 6 + 2 x^{2} + 4 x$

$- 2 x + 6 - 2 x^{2} = - 6$

$y = - 6 + 2 x^{2} + 4 x$

Step 3

Solve for $x$ in $- 2 x + 6 - 2 x^{2} = - 6$ .

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Step 3.1

Add $6$ to both sides of the equation.

$- 2 x + 6 - 2 x^{2} + 6 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.2

Add $6$ and $6$ .

$- 2 x - 2 x^{2} + 12 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $- 2$ out of $- 2 x - 2 x^{2} + 12$ .

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Step 3.3.1.1

Reorder $- 2 x$ and $- 2 x^{2}$ .

$- 2 x^{2} - 2 x + 12 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.3.1.2

Factor $- 2$ out of $- 2 x^{2}$ .

$- 2 x^{2} - 2 x + 12 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.3.1.3

Factor $- 2$ out of $- 2 x$ .

$- 2 x^{2} - 2 x + 12 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.3.1.4

Factor $- 2$ out of $12$ .

$- 2 x^{2} - 2 x - 2 \cdot - 6 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.3.1.5

Factor $- 2$ out of $- 2 (x^{2}) - 2 (x)$ .

$- 2 (x^{2} + x) - 2 \cdot - 6 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.3.1.6

Factor $- 2$ out of $- 2 (x^{2} + x) - 2 (- 6)$ .

$- 2 (x^{2} + x - 6) = 0$

$y = - 6 + 2 x^{2} + 4 x$

$- 2 (x^{2} + x - 6) = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.3.2

Factor.

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Step 3.3.2.1

Factor $x^{2} + x - 6$ using the AC method.

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Step 3.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.3.2.1.2

Write the factored form using these integers.

$- 2 ((x - 2) (x + 3)) = 0$

$y = - 6 + 2 x^{2} + 4 x$

$- 2 ((x - 2) (x + 3)) = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.3.2.2

Remove unnecessary parentheses.

$- 2 (x - 2) (x + 3) = 0$

$y = - 6 + 2 x^{2} + 4 x$

$- 2 (x - 2) (x + 3) = 0$

$y = - 6 + 2 x^{2} + 4 x$

$- 2 (x - 2) (x + 3) = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 3 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.5.2

Add $2$ to both sides of the equation.

$x = 2$

$y = - 6 + 2 x^{2} + 4 x$

$x = 2$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.6

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

$y = - 6 + 2 x^{2} + 4 x$

$x = - 3$

$y = - 6 + 2 x^{2} + 4 x$

Step 3.7

The final solution is all the values that make $- 2 (x - 2) (x + 3) = 0$ true.

$x = 2, - 3$

$y = - 6 + 2 x^{2} + 4 x$

$x = 2, - 3$

$y = - 6 + 2 x^{2} + 4 x$

Step 4

Replace all occurrences of $x$ with $2$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - 6 + 2 x^{2} + 4 x$ with $2$ .

$y = - 6 + 2 {(2)}^{2} + 4 (2)$

$x = 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 6 + 2 {(2)}^{2} + 4 (2)$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

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Step 4.2.1.1.1.1

Multiply $2$ by ${(2)}^{2}$ .

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Step 4.2.1.1.1.1.1

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

$y = - 6 + 2 {(2)}^{2} + 4 (2)$

$x = 2$

Step 4.2.1.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = - 6 + 2^{1 + 2} + 4 (2)$

$x = 2$

$y = - 6 + 2^{1 + 2} + 4 (2)$

$x = 2$

Step 4.2.1.1.1.2

Add $1$ and $2$ .

$y = - 6 + 2^{3} + 4 (2)$

$x = 2$

$y = - 6 + 2^{3} + 4 (2)$

$x = 2$

Step 4.2.1.1.2

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

$y = - 6 + 8 + 4 (2)$

$x = 2$

Step 4.2.1.1.3

Multiply $4$ by $2$ .

$y = - 6 + 8 + 8$

$x = 2$

$y = - 6 + 8 + 8$

$x = 2$

Step 4.2.1.2

Simplify by adding numbers.

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Step 4.2.1.2.1

Add $- 6$ and $8$ .

$y = 2 + 8$

$x = 2$

Step 4.2.1.2.2

Add $2$ and $8$ .

$y = 10$

$x = 2$

$y = 10$

$x = 2$

$y = 10$

$x = 2$

$y = 10$

$x = 2$

$y = 10$

$x = 2$

Step 5

Replace all occurrences of $x$ with $- 3$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - 6 + 2 x^{2} + 4 x$ with $- 3$ .

$y = - 6 + 2 {(- 3)}^{2} + 4 (- 3)$

$x = - 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 6 + 2 {(- 3)}^{2} + 4 (- 3)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

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

$y = - 6 + 2 \cdot 9 + 4 (- 3)$

$x = - 3$

Step 5.2.1.1.2

Multiply $2$ by $9$ .

$y = - 6 + 18 + 4 (- 3)$

$x = - 3$

Step 5.2.1.1.3

Multiply $4$ by $- 3$ .

$y = - 6 + 18 - 12$

$x = - 3$

$y = - 6 + 18 - 12$

$x = - 3$

Step 5.2.1.2

Simplify by adding and subtracting.

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Step 5.2.1.2.1

Add $- 6$ and $18$ .

$y = 12 - 12$

$x = - 3$

Step 5.2.1.2.2

Subtract $12$ from $12$ .

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

$y = 0$

$x = - 3$

Step 6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(2, 10)$

$(- 3, 0)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(2, 10), (- 3, 0)$

Equation Form:

$x = 2, y = 10$

$x = - 3, y = 0$

Step 8