Find the Roots (Zeros) f(x)=(2x+5)^2-32

$f (x) = {(2 x + 5)}^{2} - 32$

Step 1

Set ${(2 x + 5)}^{2} - 32$ equal to $0$ .

${(2 x + 5)}^{2} - 32 = 0$

Step 2

Solve for $x$ .

Add $32$ to both sides of the equation.

${(2 x + 5)}^{2} = 32$

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

$2 x + 5 = \pm \sqrt{32}$

Simplify $\pm \sqrt{32}$ .

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

Factor $16$ out of $32$ .

$2 x + 5 = \pm \sqrt{16 (2)}$

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

$2 x + 5 = \pm \sqrt{4^{2} \cdot 2}$

Pull terms out from under the radical.

$2 x + 5 = \pm 4 \sqrt{2}$

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

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

$2 x + 5 = 4 \sqrt{2}$

Subtract $5$ from both sides of the equation.

$2 x = 4 \sqrt{2} - 5$

Divide each term in $2 x = 4 \sqrt{2} - 5$ by $2$ and simplify.

Divide each term in $2 x = 4 \sqrt{2} - 5$ by $2$ .

$\frac{2 x}{2} = \frac{4 \sqrt{2}}{2} + \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4 \sqrt{2}}{2} + \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{4 \sqrt{2}}{2} + \frac{- 5}{2}$

Simplify the right side.

Simplify each term.

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

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

$x = \frac{2 (2 \sqrt{2})}{2} + \frac{- 5}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = \frac{2 (2 \sqrt{2})}{2 (1)} + \frac{- 5}{2}$

Cancel the common factor.

$x = \frac{2 (2 \sqrt{2})}{2 \cdot 1} + \frac{- 5}{2}$

Rewrite the expression.

$x = \frac{2 \sqrt{2}}{1} + \frac{- 5}{2}$

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

$x = 2 \sqrt{2} + \frac{- 5}{2}$

Move the negative in front of the fraction.

$x = 2 \sqrt{2} - \frac{5}{2}$

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

$2 x + 5 = - 4 \sqrt{2}$

Subtract $5$ from both sides of the equation.

$2 x = - 4 \sqrt{2} - 5$

Divide each term in $2 x = - 4 \sqrt{2} - 5$ by $2$ and simplify.

Divide each term in $2 x = - 4 \sqrt{2} - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 4 \sqrt{2}}{2} + \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 4 \sqrt{2}}{2} + \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{- 4 \sqrt{2}}{2} + \frac{- 5}{2}$

Simplify the right side.

Simplify each term.

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

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

$x = \frac{2 (- 2 \sqrt{2})}{2} + \frac{- 5}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = \frac{2 (- 2 \sqrt{2})}{2 (1)} + \frac{- 5}{2}$

Cancel the common factor.

$x = \frac{2 (- 2 \sqrt{2})}{2 \cdot 1} + \frac{- 5}{2}$

Rewrite the expression.

$x = \frac{- 2 \sqrt{2}}{1} + \frac{- 5}{2}$

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

$x = - 2 \sqrt{2} + \frac{- 5}{2}$

Move the negative in front of the fraction.

$x = - 2 \sqrt{2} - \frac{5}{2}$

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

$x = 2 \sqrt{2} - \frac{5}{2}, - 2 \sqrt{2} - \frac{5}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 2 \sqrt{2} - \frac{5}{2}, - 2 \sqrt{2} - \frac{5}{2}$

Decimal Form:

$x = 0.32842712 \dots, - 5.32842712 \dots$

Step 4

Find the Roots (Zeros) f(x)=(2x+5)^2-32

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