Precalculus Examples

$8 = 2 {(3 x + 3)}^{2}$ , $(- 1, 3)$

Step 1

Rewrite the equation as $2 {(3 x + 3)}^{2} = 8$ .

$2 {(3 x + 3)}^{2} = 8$

Step 2

Divide each term in $2 {(3 x + 3)}^{2} = 8$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 {(3 x + 3)}^{2} = 8$ by $2$ .

$\frac{2 {(3 x + 3)}^{2}}{2} = \frac{8}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 {(3 x + 3)}^{2}}{2} = \frac{8}{2}$

Step 2.2.1.2

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

${(3 x + 3)}^{2} = \frac{8}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $8$ by $2$ .

${(3 x + 3)}^{2} = 4$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$3 x + 3 = \pm \sqrt{4}$

Step 4

Simplify $\pm \sqrt{4}$ .

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Step 4.1

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

$3 x + 3 = \pm \sqrt{2^{2}}$

Step 4.2

Pull terms out from under the radical, assuming positive real numbers.

$3 x + 3 = \pm 2$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$3 x + 3 = 2$

Step 5.2

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

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Step 5.2.1

Subtract $3$ from both sides of the equation.

$3 x = 2 - 3$

Step 5.2.2

Subtract $3$ from $2$ .

$3 x = - 1$

Step 5.3

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 5.3.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 5.4

Next, use the negative value of the $\pm$ to find the second solution.

$3 x + 3 = - 2$

Step 5.5

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

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Step 5.5.1

Subtract $3$ from both sides of the equation.

$3 x = - 2 - 3$

Step 5.5.2

Subtract $3$ from $- 2$ .

$3 x = - 5$

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $3$ .

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Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Divide $x$ by $1$ .

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

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Move the negative in front of the fraction.

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

Step 5.7

The complete solution is the result of both the positive and negative portions of the solution.

$x = - \frac{1}{3}, - \frac{5}{3}$

Step 6

Find the values of $n$ that produce a value within the interval $(- 1, 3)$ .

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Step 6.1

The interval $(- 1, 3)$ does not contain $- \frac{5}{3}$ . It is not part of the final solution.

$- \frac{5}{3}$ is not on the interval

Step 6.2

The interval $(- 1, 3)$ contains $- \frac{1}{3}$ .

$x = - \frac{1}{3}$

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