Precalculus Examples

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

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

(- 1, 3)

$(- 1, 3)$

Step 1

Rewrite the equation as

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

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

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

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

Step 2

Divide each term in

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

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

2

$2$ and simplify.

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Step 2.1

Divide each term in

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

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

2

$2$ .

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

$2$ .

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Step 2.2.1.1

Cancel the common factor.

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

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

Step 2.2.1.2

Divide

{(3 x + 3)}^{2}

${(3 x + 3)}^{2}$ by

1

$1$ .

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

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

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

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

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

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide

8

$8$ by

2

$2$ .

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

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

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

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

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

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

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}

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

Step 4

Simplify

\pm \sqrt{4}

$\pm \sqrt{4}$ .

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Step 4.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

3 x + 3 = \pm \sqrt{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

$3 x + 3 = \pm 2$

3 x + 3 = \pm 2

$3 x + 3 = \pm 2$

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

$\pm$ to find the first solution.

3 x + 3 = 2

$3 x + 3 = 2$

Step 5.2

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 5.2.1

Subtract

3

$3$ from both sides of the equation.

3 x = 2 - 3

$3 x = 2 - 3$

Step 5.2.2

Subtract

3

$3$ from

2

$2$ .

3 x = - 1

$3 x = - 1$

3 x = - 1

$3 x = - 1$

Step 5.3

Divide each term in

3 x = - 1

$3 x = - 1$ by

3

$3$ and simplify.

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Step 5.3.1

Divide each term in

3 x = - 1

$3 x = - 1$ by

3

$3$ .

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

$3$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

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

Step 5.3.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 1}{3}

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

x = \frac{- 1}{3}

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

x = \frac{- 1}{3}

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

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}

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

x = - \frac{1}{3}

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

x = - \frac{1}{3}

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

Step 5.4

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

3 x + 3 = - 2

$3 x + 3 = - 2$

Step 5.5

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 5.5.1

Subtract

3

$3$ from both sides of the equation.

3 x = - 2 - 3

$3 x = - 2 - 3$

Step 5.5.2

Subtract

3

$3$ from

- 2

$- 2$ .

3 x = - 5

$3 x = - 5$

3 x = - 5

$3 x = - 5$

Step 5.6

Divide each term in

3 x = - 5

$3 x = - 5$ by

3

$3$ and simplify.

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Step 5.6.1

Divide each term in

3 x = - 5

$3 x = - 5$ by

3

$3$ .

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

$3$ .

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Step 5.6.2.1.1

Cancel the common factor.

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

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

Step 5.6.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 5}{3}

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

x = \frac{- 5}{3}

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

x = \frac{- 5}{3}

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

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}

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

x = - \frac{5}{3}

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

x = - \frac{5}{3}

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

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}

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

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

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

Step 6

Find the values of

n

$n$ that produce a value within the interval

(- 1, 3)

$(- 1, 3)$ .

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Step 6.1

The interval

(- 1, 3)

$(- 1, 3)$ does not contain

- \frac{5}{3}

$- \frac{5}{3}$ . It is not part of the final solution.

- \frac{5}{3}

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

Step 6.2

The interval

(- 1, 3)

$(- 1, 3)$ contains

- \frac{1}{3}

$- \frac{1}{3}$ .

x = - \frac{1}{3}

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

x = - \frac{1}{3}

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

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