Pre-Algebra Examples

$36 = \frac{1}{2} \cdot ((x + 2) (x + 8))$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot ((x + 2) (x + 8)) = 36$ .

$\frac{1}{2} \cdot ((x + 2) (x + 8)) = 36$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot ((x + 2) (x + 8))) = 2 \cdot 36$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $2 (\frac{1}{2} \cdot ((x + 2) (x + 8)))$ .

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Step 3.1.1.1

Expand $(x + 2) (x + 8)$ using the FOIL Method.

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Step 3.1.1.1.1

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x (x + 8) + 2 (x + 8))) = 2 \cdot 36$

Step 3.1.1.1.2

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x \cdot x + x \cdot 8 + 2 (x + 8))) = 2 \cdot 36$

Step 3.1.1.1.3

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x \cdot x + x \cdot 8 + 2 x + 2 \cdot 8)) = 2 \cdot 36$

Step 3.1.1.2

Simplify and combine like terms.

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Step 3.1.1.2.1

Simplify each term.

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Step 3.1.1.2.1.1

Multiply $x$ by $x$ .

$2 (\frac{1}{2} \cdot (x^{2} + x \cdot 8 + 2 x + 2 \cdot 8)) = 2 \cdot 36$

Step 3.1.1.2.1.2

Move $8$ to the left of $x$ .

$2 (\frac{1}{2} \cdot (x^{2} + 8 \cdot x + 2 x + 2 \cdot 8)) = 2 \cdot 36$

Step 3.1.1.2.1.3

Multiply $2$ by $8$ .

$2 (\frac{1}{2} \cdot (x^{2} + 8 x + 2 x + 16)) = 2 \cdot 36$

Step 3.1.1.2.2

Add $8 x$ and $2 x$ .

$2 (\frac{1}{2} \cdot (x^{2} + 10 x + 16)) = 2 \cdot 36$

Step 3.1.1.3

Apply the distributive property.

$2 (\frac{1}{2} x^{2} + \frac{1}{2} (10 x) + \frac{1}{2} \cdot 16) = 2 \cdot 36$

Step 3.1.1.4

Simplify.

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Step 3.1.1.4.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$2 (\frac{x^{2}}{2} + \frac{1}{2} (10 x) + \frac{1}{2} \cdot 16) = 2 \cdot 36$

Step 3.1.1.4.2

Cancel the common factor of $2$ .

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Step 3.1.1.4.2.1

Factor $2$ out of $10 x$ .

$2 (\frac{x^{2}}{2} + \frac{1}{2} (2 (5 x)) + \frac{1}{2} \cdot 16) = 2 \cdot 36$

Step 3.1.1.4.2.2

Cancel the common factor.

$2 (\frac{x^{2}}{2} + \frac{1}{2} (2 (5 x)) + \frac{1}{2} \cdot 16) = 2 \cdot 36$

Step 3.1.1.4.2.3

Rewrite the expression.

$2 (\frac{x^{2}}{2} + 5 x + \frac{1}{2} \cdot 16) = 2 \cdot 36$

Step 3.1.1.4.3

Cancel the common factor of $2$ .

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Step 3.1.1.4.3.1

Factor $2$ out of $16$ .

$2 (\frac{x^{2}}{2} + 5 x + \frac{1}{2} \cdot (2 (8))) = 2 \cdot 36$

Step 3.1.1.4.3.2

Cancel the common factor.

$2 (\frac{x^{2}}{2} + 5 x + \frac{1}{2} \cdot (2 \cdot 8)) = 2 \cdot 36$

Step 3.1.1.4.3.3

Rewrite the expression.

$2 (\frac{x^{2}}{2} + 5 x + 8) = 2 \cdot 36$

Step 3.1.1.5

Apply the distributive property.

$2 \frac{x^{2}}{2} + 2 (5 x) + 2 \cdot 8 = 2 \cdot 36$

Step 3.1.1.6

Simplify.

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Step 3.1.1.6.1

Cancel the common factor of $2$ .

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Step 3.1.1.6.1.1

Cancel the common factor.

$2 \frac{x^{2}}{2} + 2 (5 x) + 2 \cdot 8 = 2 \cdot 36$

Step 3.1.1.6.1.2

Rewrite the expression.

$x^{2} + 2 (5 x) + 2 \cdot 8 = 2 \cdot 36$

Step 3.1.1.6.2

Multiply $5$ by $2$ .

$x^{2} + 10 x + 2 \cdot 8 = 2 \cdot 36$

Step 3.1.1.6.3

Multiply $2$ by $8$ .

$x^{2} + 10 x + 16 = 2 \cdot 36$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $36$ .

$x^{2} + 10 x + 16 = 72$

Step 4

Subtract $72$ from both sides of the equation.

$x^{2} + 10 x + 16 - 72 = 0$

Step 5

Subtract $72$ from $16$ .

$x^{2} + 10 x - 56 = 0$

Step 6

Factor $x^{2} + 10 x - 56$ using the AC method.

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Step 6.1

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 $- 56$ and whose sum is $10$ .

$- 4, 14$

Step 6.2

Write the factored form using these integers.

$(x - 4) (x + 14) = 0$

Step 7

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

$x - 4 = 0$

$x + 14 = 0$

Step 8

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 8.2

Add $4$ to both sides of the equation.

$x = 4$

Step 9

Set $x + 14$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x + 14$ equal to $0$ .

$x + 14 = 0$

Step 9.2

Subtract $14$ from both sides of the equation.

$x = - 14$

Step 10

The final solution is all the values that make $(x - 4) (x + 14) = 0$ true.

$x = 4, - 14$