Algebra Examples

$y = 2 x^{2} + x - 108$ $3 x + y = 18$

Step 1

Replace all occurrences of $y$ with $2 x^{2} + x - 108$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $3 x + y = 18$ with $2 x^{2} + x - 108$ .

$3 x + 2 x^{2} + x - 108 = 18$

$y = 2 x^{2} + x - 108$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $3 x + 2 x^{2} + x - 108$ .

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Step 1.2.1.1

Remove parentheses.

$3 x + 2 x^{2} + x - 108 = 18$

$y = 2 x^{2} + x - 108$

Step 1.2.1.2

Add $3 x$ and $x$ .

$2 x^{2} + 4 x - 108 = 18$

$y = 2 x^{2} + x - 108$

$2 x^{2} + 4 x - 108 = 18$

$y = 2 x^{2} + x - 108$

$2 x^{2} + 4 x - 108 = 18$

$y = 2 x^{2} + x - 108$

$2 x^{2} + 4 x - 108 = 18$

$y = 2 x^{2} + x - 108$

Step 2

Solve for $x$ in $2 x^{2} + 4 x - 108 = 18$ .

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Step 2.1

Subtract $18$ from both sides of the equation.

$2 x^{2} + 4 x - 108 - 18 = 0$

$y = 2 x^{2} + x - 108$

Step 2.2

Subtract $18$ from $- 108$ .

$2 x^{2} + 4 x - 126 = 0$

$y = 2 x^{2} + x - 108$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor $2$ out of $2 x^{2} + 4 x - 126$ .

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Step 2.3.1.1

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

$2 (x^{2}) + 4 x - 126 = 0$

$y = 2 x^{2} + x - 108$

Step 2.3.1.2

Factor $2$ out of $4 x$ .

$2 (x^{2}) + 2 (2 x) - 126 = 0$

$y = 2 x^{2} + x - 108$

Step 2.3.1.3

Factor $2$ out of $- 126$ .

$2 x^{2} + 2 (2 x) + 2 \cdot - 63 = 0$

$y = 2 x^{2} + x - 108$

Step 2.3.1.4

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

$2 (x^{2} + 2 x) + 2 \cdot - 63 = 0$

$y = 2 x^{2} + x - 108$

Step 2.3.1.5

Factor $2$ out of $2 (x^{2} + 2 x) + 2 \cdot - 63$ .

$2 (x^{2} + 2 x - 63) = 0$

$y = 2 x^{2} + x - 108$

$2 (x^{2} + 2 x - 63) = 0$

$y = 2 x^{2} + x - 108$

Step 2.3.2

Factor.

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Step 2.3.2.1

Factor $x^{2} + 2 x - 63$ using the AC method.

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Step 2.3.2.1.1

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 $- 63$ and whose sum is $2$ .

$- 7, 9$

$y = 2 x^{2} + x - 108$

Step 2.3.2.1.2

Write the factored form using these integers.

$2 ((x - 7) (x + 9)) = 0$

$y = 2 x^{2} + x - 108$

$2 ((x - 7) (x + 9)) = 0$

$y = 2 x^{2} + x - 108$

Step 2.3.2.2

Remove unnecessary parentheses.

$2 (x - 7) (x + 9) = 0$

$y = 2 x^{2} + x - 108$

$2 (x - 7) (x + 9) = 0$

$y = 2 x^{2} + x - 108$

$2 (x - 7) (x + 9) = 0$

$y = 2 x^{2} + x - 108$

Step 2.4

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

$x - 7 = 0$

$x + 9 = 0$

$y = 2 x^{2} + x - 108$

Step 2.5

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

$y = 2 x^{2} + x - 108$

Step 2.5.2

Add $7$ to both sides of the equation.

$x = 7$

$y = 2 x^{2} + x - 108$

$x = 7$

$y = 2 x^{2} + x - 108$

Step 2.6

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

$y = 2 x^{2} + x - 108$

Step 2.6.2

Subtract $9$ from both sides of the equation.

$x = - 9$

$y = 2 x^{2} + x - 108$

$x = - 9$

$y = 2 x^{2} + x - 108$

Step 2.7

The final solution is all the values that make $2 (x - 7) (x + 9) = 0$ true.

$x = 7, - 9$

$y = 2 x^{2} + x - 108$

$x = 7, - 9$

$y = 2 x^{2} + x - 108$

Step 3

Replace all occurrences of $x$ with $7$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $y = 2 x^{2} + x - 108$ with $7$ .

$y = 2 {(7)}^{2} + 7 - 108$

$x = 7$

Step 3.2

Simplify $y = 2 {(7)}^{2} + 7 - 108$ .

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Remove parentheses.

$y = 2 {(7)}^{2} + 7 - 108$

$x = 7$

$y = 2 {(7)}^{2} + 7 - 108$

$x = 7$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 {(7)}^{2} + 7 - 108$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Raise $7$ to the power of $2$ .

$y = 2 \cdot 49 + 7 - 108$

$x = 7$

Step 3.2.2.1.1.2

Multiply $2$ by $49$ .

$y = 98 + 7 - 108$

$x = 7$

$y = 98 + 7 - 108$

$x = 7$

Step 3.2.2.1.2

Simplify by adding and subtracting.

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Step 3.2.2.1.2.1

Add $98$ and $7$ .

$y = 105 - 108$

$x = 7$

Step 3.2.2.1.2.2

Subtract $108$ from $105$ .

$y = - 3$

$x = 7$

$y = - 3$

$x = 7$

$y = - 3$

$x = 7$

$y = - 3$

$x = 7$

$y = - 3$

$x = 7$

$y = - 3$

$x = 7$

Step 4

Replace all occurrences of $x$ with $- 9$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 2 x^{2} + x - 108$ with $- 9$ .

$y = 2 {(- 9)}^{2} - 9 - 108$

$x = - 9$

Step 4.2

Simplify $y = 2 {(- 9)}^{2} - 9 - 108$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$y = 2 {(- 9)}^{2} - 9 - 108$

$x = - 9$

$y = 2 {(- 9)}^{2} - 9 - 108$

$x = - 9$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $2 {(- 9)}^{2} - 9 - 108$ .

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Step 4.2.2.1.1

Simplify each term.

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Step 4.2.2.1.1.1

Raise $- 9$ to the power of $2$ .

$y = 2 \cdot 81 - 9 - 108$

$x = - 9$

Step 4.2.2.1.1.2

Multiply $2$ by $81$ .

$y = 162 - 9 - 108$

$x = - 9$

$y = 162 - 9 - 108$

$x = - 9$

Step 4.2.2.1.2

Simplify by subtracting numbers.

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Step 4.2.2.1.2.1

Subtract $9$ from $162$ .

$y = 153 - 108$

$x = - 9$

Step 4.2.2.1.2.2

Subtract $108$ from $153$ .

$y = 45$

$x = - 9$

$y = 45$

$x = - 9$

$y = 45$

$x = - 9$

$y = 45$

$x = - 9$

$y = 45$

$x = - 9$

$y = 45$

$x = - 9$

Step 5

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

$(7, - 3)$

$(- 9, 45)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(7, - 3), (- 9, 45)$

Equation Form:

$x = 7, y = - 3$

$x = - 9, y = 45$

Step 7