Pre-Algebra Examples

${(6 x + 6)}^{2} + 8 (6 x + 6) + 16 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = 6 x + 6$ . Substitute $u$ for all occurrences of $6 x + 6$ .

$u^{2} + 8 u + 16 = 0$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $16$ as $4^{2}$ .

$u^{2} + 8 u + 4^{2} = 0$

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 u = 2 \cdot u \cdot 4$

Step 1.2.3

Rewrite the polynomial.

$u^{2} + 2 \cdot u \cdot 4 + 4^{2} = 0$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 4$ .

${(u + 4)}^{2} = 0$

Step 1.3

Replace all occurrences of $u$ with $6 x + 6$ .

${(6 x + 6 + 4)}^{2} = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Add $6$ and $4$ .

${(6 x + 10)}^{2} = 0$

Step 2.2

Factor $2$ out of $6 x + 10$ .

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Step 2.2.1

Factor $2$ out of $6 x$ .

${(2 (3 x) + 10)}^{2} = 0$

Step 2.2.2

Factor $2$ out of $10$ .

${(2 (3 x) + 2 (5))}^{2} = 0$

Step 2.2.3

Factor $2$ out of $2 (3 x) + 2 (5)$ .

${(2 (3 x + 5))}^{2} = 0$

Step 2.3

Apply the product rule to $2 (3 x + 5)$ .

$2^{2} {(3 x + 5)}^{2} = 0$

Step 3

Divide each term in $2^{2} {(3 x + 5)}^{2} = 0$ by $2^{2}$ and simplify.

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Step 3.1

Divide each term in $2^{2} {(3 x + 5)}^{2} = 0$ by $2^{2}$ .

$\frac{2^{2} {(3 x + 5)}^{2}}{2^{2}} = \frac{0}{2^{2}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2^{2} {(3 x + 5)}^{2}}{2^{2}} = \frac{0}{2^{2}}$

Step 3.2.1.2

Divide ${(3 x + 5)}^{2}$ by $1$ .

${(3 x + 5)}^{2} = \frac{0}{2^{2}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

${(3 x + 5)}^{2} = \frac{0}{4}$

Step 3.3.2

Divide $0$ by $4$ .

${(3 x + 5)}^{2} = 0$

Step 4

Set the $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Step 5.2

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{3}$