Finite Math Examples

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

Step 1

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

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Step 1.1

Add $x^{2}$ to both sides of the equation.

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

$y + x = 2$

Step 1.2

Subtract $2 x$ from both sides of the equation.

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

$y + x = 2$

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

$y + x = 2$

Step 2

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

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Step 2.1

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Add $- 2 x$ and $x$ .

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

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

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

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

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

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

Step 3

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

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Step 3.1

Subtract $2$ from both sides of the equation.

$- 4 + x^{2} - x - 2 = 0$

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

Step 3.2

Subtract $2$ from $- 4$ .

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

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

Step 3.3

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

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Step 3.3.1

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 $- 1$ .

$- 3, 2$

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

Step 3.3.2

Write the factored form using these integers.

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

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

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

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

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 $0$ .

$x - 3 = 0$

$x + 2 = 0$

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

Step 3.5

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

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Step 3.5.1

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

$x - 3 = 0$

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

Step 3.5.2

Add $3$ to both sides of the equation.

$x = 3$

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

$x = 3$

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

Step 3.6

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

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Step 3.6.1

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

$x + 2 = 0$

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

Step 3.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

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

$x = - 2$

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

Step 3.7

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

$x = 3, - 2$

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

$x = 3, - 2$

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

Step 4

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

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Step 4.1

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

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

$x = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

$y = - 4 + 9 - 2 \cdot 3$

$x = 3$

Step 4.2.1.1.2

Multiply $- 2$ by $3$ .

$y = - 4 + 9 - 6$

$x = 3$

$y = - 4 + 9 - 6$

$x = 3$

Step 4.2.1.2

Simplify by adding and subtracting.

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Step 4.2.1.2.1

Add $- 4$ and $9$ .

$y = 5 - 6$

$x = 3$

Step 4.2.1.2.2

Subtract $6$ from $5$ .

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

Step 5

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

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Step 5.1

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

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

$x = - 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

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

$y = - 4 + 4 - 2 \cdot - 2$

$x = - 2$

Step 5.2.1.1.2

Multiply $- 2$ by $- 2$ .

$y = - 4 + 4 + 4$

$x = - 2$

$y = - 4 + 4 + 4$

$x = - 2$

Step 5.2.1.2

Simplify by adding numbers.

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Step 5.2.1.2.1

Add $- 4$ and $4$ .

$y = 0 + 4$

$x = - 2$

Step 5.2.1.2.2

Add $0$ and $4$ .

$y = 4$

$x = - 2$

$y = 4$

$x = - 2$

$y = 4$

$x = - 2$

$y = 4$

$x = - 2$

$y = 4$

$x = - 2$

Step 6

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

$(3, - 1)$

$(- 2, 4)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(3, - 1), (- 2, 4)$

Equation Form:

$x = 3, y = - 1$

$x = - 2, y = 4$

Step 8