Pre-Algebra Examples

$\frac{x (x + 6)}{2} = 108$

Step 1

Multiply both sides of the equation by $2$ .

$2 \frac{x (x + 6)}{2} = 2 \cdot 108$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $2 \frac{x (x + 6)}{2}$ .

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Step 2.1.1.1

Cancel the common factor of $2$ .

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Step 2.1.1.1.1

Cancel the common factor.

$2 \frac{x (x + 6)}{2} = 2 \cdot 108$

Step 2.1.1.1.2

Rewrite the expression.

$x (x + 6) = 2 \cdot 108$

Step 2.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 6 = 2 \cdot 108$

Step 2.1.1.3

Simplify the expression.

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Step 2.1.1.3.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 6 = 2 \cdot 108$

Step 2.1.1.3.2

Move $6$ to the left of $x$ .

$x^{2} + 6 x = 2 \cdot 108$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $2$ by $108$ .

$x^{2} + 6 x = 216$

Step 3

Subtract $216$ from both sides of the equation.

$x^{2} + 6 x - 216 = 0$

Step 4

Factor $x^{2} + 6 x - 216$ using the AC method.

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Step 4.1

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 $- 216$ and whose sum is $6$ .

$- 12, 18$

Step 4.2

Write the factored form using these integers.

$(x - 12) (x + 18) = 0$

Step 5

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

$x - 12 = 0$

$x + 18 = 0$

Step 6

Set $x - 12$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Step 6.2

Add $12$ to both sides of the equation.

$x = 12$

Step 7

Set $x + 18$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x + 18$ equal to $0$ .

$x + 18 = 0$

Step 7.2

Subtract $18$ from both sides of the equation.

$x = - 18$

Step 8

The final solution is all the values that make $(x - 12) (x + 18) = 0$ true.

$x = 12, - 18$