Algebra Examples

${(2 x + 9)}^{2} - 32$

Step 1

Set ${(2 x + 9)}^{2} - 32$ equal to $0$ .

${(2 x + 9)}^{2} - 32 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $32$ to both sides of the equation.

${(2 x + 9)}^{2} = 32$

Step 2.2

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

$2 x + 9 = \pm \sqrt{32}$

Step 2.3

Simplify $\pm \sqrt{32}$ .

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Step 2.3.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.3.1.1

Factor $16$ out of $32$ .

$2 x + 9 = \pm \sqrt{16 (2)}$

Step 2.3.1.2

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

$2 x + 9 = \pm \sqrt{4^{2} \cdot 2}$

Step 2.3.2

Pull terms out from under the radical.

$2 x + 9 = \pm 4 \sqrt{2}$

Step 2.4

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

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Step 2.4.1

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

$2 x + 9 = 4 \sqrt{2}$

Step 2.4.2

Subtract $9$ from both sides of the equation.

$2 x = 4 \sqrt{2} - 9$

Step 2.4.3

Divide each term in $2 x = 4 \sqrt{2} - 9$ by $2$ and simplify.

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Step 2.4.3.1

Divide each term in $2 x = 4 \sqrt{2} - 9$ by $2$ .

$\frac{2 x}{2} = \frac{4 \sqrt{2}}{2} + \frac{- 9}{2}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $2$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4 \sqrt{2}}{2} + \frac{- 9}{2}$

Step 2.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4 \sqrt{2}}{2} + \frac{- 9}{2}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Simplify each term.

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Step 2.4.3.3.1.1

Cancel the common factor of $4$ and $2$ .

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Step 2.4.3.3.1.1.1

Factor $2$ out of $4 \sqrt{2}$ .

$x = \frac{2 (2 \sqrt{2})}{2} + \frac{- 9}{2}$

Step 2.4.3.3.1.1.2

Cancel the common factors.

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Step 2.4.3.3.1.1.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (2 \sqrt{2})}{2 (1)} + \frac{- 9}{2}$

Step 2.4.3.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 (2 \sqrt{2})}{2 \cdot 1} + \frac{- 9}{2}$

Step 2.4.3.3.1.1.2.3

Rewrite the expression.

$x = \frac{2 \sqrt{2}}{1} + \frac{- 9}{2}$

Step 2.4.3.3.1.1.2.4

Divide $2 \sqrt{2}$ by $1$ .

$x = 2 \sqrt{2} + \frac{- 9}{2}$

Step 2.4.3.3.1.2

Move the negative in front of the fraction.

$x = 2 \sqrt{2} - \frac{9}{2}$

Step 2.4.4

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

$2 x + 9 = - 4 \sqrt{2}$

Step 2.4.5

Subtract $9$ from both sides of the equation.

$2 x = - 4 \sqrt{2} - 9$

Step 2.4.6

Divide each term in $2 x = - 4 \sqrt{2} - 9$ by $2$ and simplify.

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Step 2.4.6.1

Divide each term in $2 x = - 4 \sqrt{2} - 9$ by $2$ .

$\frac{2 x}{2} = \frac{- 4 \sqrt{2}}{2} + \frac{- 9}{2}$

Step 2.4.6.2

Simplify the left side.

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Step 2.4.6.2.1

Cancel the common factor of $2$ .

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Step 2.4.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 4 \sqrt{2}}{2} + \frac{- 9}{2}$

Step 2.4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4 \sqrt{2}}{2} + \frac{- 9}{2}$

Step 2.4.6.3

Simplify the right side.

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Step 2.4.6.3.1

Simplify each term.

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Step 2.4.6.3.1.1

Cancel the common factor of $- 4$ and $2$ .

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Step 2.4.6.3.1.1.1

Factor $2$ out of $- 4 \sqrt{2}$ .

$x = \frac{2 (- 2 \sqrt{2})}{2} + \frac{- 9}{2}$

Step 2.4.6.3.1.1.2

Cancel the common factors.

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Step 2.4.6.3.1.1.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (- 2 \sqrt{2})}{2 (1)} + \frac{- 9}{2}$

Step 2.4.6.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 (- 2 \sqrt{2})}{2 \cdot 1} + \frac{- 9}{2}$

Step 2.4.6.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 2 \sqrt{2}}{1} + \frac{- 9}{2}$

Step 2.4.6.3.1.1.2.4

Divide $- 2 \sqrt{2}$ by $1$ .

$x = - 2 \sqrt{2} + \frac{- 9}{2}$

Step 2.4.6.3.1.2

Move the negative in front of the fraction.

$x = - 2 \sqrt{2} - \frac{9}{2}$

Step 2.4.7

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

$x = 2 \sqrt{2} - \frac{9}{2}, - 2 \sqrt{2} - \frac{9}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 2 \sqrt{2} - \frac{9}{2}, - 2 \sqrt{2} - \frac{9}{2}$

Decimal Form:

$x = - 1.67157287 \dots, - 7.32842712 \dots$

Step 4