Basic Math Examples

${(2 b - 7)}^{2} = 32$

Step 1

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

$2 b - 7 = \pm \sqrt{32}$

Step 2

Simplify $\pm \sqrt{32}$ .

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Step 2.1

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

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Step 2.1.1

Factor $16$ out of $32$ .

$2 b - 7 = \pm \sqrt{16 (2)}$

Step 2.1.2

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

$2 b - 7 = \pm \sqrt{4^{2} \cdot 2}$

Step 2.2

Pull terms out from under the radical.

$2 b - 7 = \pm 4 \sqrt{2}$

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.

$2 b - 7 = 4 \sqrt{2}$

Step 3.2

Add $7$ to both sides of the equation.

$2 b = 4 \sqrt{2} + 7$

Step 3.3

Divide each term in $2 b = 4 \sqrt{2} + 7$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 b = 4 \sqrt{2} + 7$ by $2$ .

$\frac{2 b}{2} = \frac{4 \sqrt{2}}{2} + \frac{7}{2}$

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.

$\frac{2 b}{2} = \frac{4 \sqrt{2}}{2} + \frac{7}{2}$

Step 3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{4 \sqrt{2}}{2} + \frac{7}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

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

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Step 3.3.3.1.1

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

$b = \frac{2 (2 \sqrt{2})}{2} + \frac{7}{2}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $2$ out of $2$ .

$b = \frac{2 (2 \sqrt{2})}{2 (1)} + \frac{7}{2}$

Step 3.3.3.1.2.2

Cancel the common factor.

$b = \frac{2 (2 \sqrt{2})}{2 \cdot 1} + \frac{7}{2}$

Step 3.3.3.1.2.3

Rewrite the expression.

$b = \frac{2 \sqrt{2}}{1} + \frac{7}{2}$

Step 3.3.3.1.2.4

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

$b = 2 \sqrt{2} + \frac{7}{2}$

Step 3.4

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

$2 b - 7 = - 4 \sqrt{2}$

Step 3.5

Add $7$ to both sides of the equation.

$2 b = - 4 \sqrt{2} + 7$

Step 3.6

Divide each term in $2 b = - 4 \sqrt{2} + 7$ by $2$ and simplify.

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Step 3.6.1

Divide each term in $2 b = - 4 \sqrt{2} + 7$ by $2$ .

$\frac{2 b}{2} = \frac{- 4 \sqrt{2}}{2} + \frac{7}{2}$

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.

$\frac{2 b}{2} = \frac{- 4 \sqrt{2}}{2} + \frac{7}{2}$

Step 3.6.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 4 \sqrt{2}}{2} + \frac{7}{2}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

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

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Step 3.6.3.1.1

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

$b = \frac{2 (- 2 \sqrt{2})}{2} + \frac{7}{2}$

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 $2$ .

$b = \frac{2 (- 2 \sqrt{2})}{2 (1)} + \frac{7}{2}$

Step 3.6.3.1.2.2

Cancel the common factor.

$b = \frac{2 (- 2 \sqrt{2})}{2 \cdot 1} + \frac{7}{2}$

Step 3.6.3.1.2.3

Rewrite the expression.

$b = \frac{- 2 \sqrt{2}}{1} + \frac{7}{2}$

Step 3.6.3.1.2.4

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

$b = - 2 \sqrt{2} + \frac{7}{2}$

Step 3.7

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

$b = 2 \sqrt{2} + \frac{7}{2}, - 2 \sqrt{2} + \frac{7}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$b = 2 \sqrt{2} + \frac{7}{2}, - 2 \sqrt{2} + \frac{7}{2}$

Decimal Form:

$b = 6.32842712 \dots, 0.67157287 \dots$