Precalculus Examples

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

Step 1

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

$2 x - 1 = \pm \sqrt{8}$

Step 2

Simplify $\pm \sqrt{8}$ .

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Step 2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1.1

Factor $4$ out of $8$ .

$2 x - 1 = \pm \sqrt{4 (2)}$

Step 2.1.2

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

$2 x - 1 = \pm \sqrt{2^{2} \cdot 2}$

Step 2.2

Pull terms out from under the radical.

$2 x - 1 = \pm 2 \sqrt{2}$

Step 3

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

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Step 3.1

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

$2 x - 1 = 2 \sqrt{2}$

Step 3.2

Add $1$ to both sides of the equation.

$2 x = 2 \sqrt{2} + 1$

Step 3.3

Divide each term in $2 x = 2 \sqrt{2} + 1$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = 2 \sqrt{2} + 1$ by $2$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $2$ .

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Step 3.3.3.1.1

Cancel the common factor.

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

Step 3.3.3.1.2

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

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

Step 3.4

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

$2 x - 1 = - 2 \sqrt{2}$

Step 3.5

Add $1$ to both sides of the equation.

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

Step 3.6

Divide each term in $2 x = - 2 \sqrt{2} + 1$ by $2$ and simplify.

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Step 3.6.1

Divide each term in $2 x = - 2 \sqrt{2} + 1$ by $2$ .

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

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.1.1

Cancel the common factor.

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

Step 3.6.2.1.2

Divide $x$ by $1$ .

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

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

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

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Step 3.6.3.1.1

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

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

Step 3.6.3.1.2

Cancel the common factors.

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Step 3.6.3.1.2.1

Factor $2$ out of $2$ .

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

Step 3.6.3.1.2.2

Cancel the common factor.

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

Step 3.6.3.1.2.3

Rewrite the expression.

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

Step 3.6.3.1.2.4

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

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

Step 3.7

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.91421356 \dots, - 0.91421356 \dots$