Solve Using the Square Root Property (4x+21)^2=45

${(4 x + 21)}^{2} = 45$

Step 1

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

$4 x + 21 = \pm \sqrt{45}$

Step 2

Simplify $\pm \sqrt{45}$ .

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$4 x + 21 = \pm \sqrt{9 (5)}$

Rewrite $9$ as $3^{2}$ .

$4 x + 21 = \pm \sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$4 x + 21 = \pm 3 \sqrt{5}$

Step 3

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.

$4 x + 21 = 3 \sqrt{5}$

Subtract $21$ from both sides of the equation.

$4 x = 3 \sqrt{5} - 21$

Divide each term in $4 x = 3 \sqrt{5} - 21$ by $4$ and simplify.

Divide each term in $4 x = 3 \sqrt{5} - 21$ by $4$ .

$\frac{4 x}{4} = \frac{3 \sqrt{5}}{4} + \frac{- 21}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{3 \sqrt{5}}{4} + \frac{- 21}{4}$

Divide $x$ by $1$ .

$x = \frac{3 \sqrt{5}}{4} + \frac{- 21}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{3 \sqrt{5}}{4} - \frac{21}{4}$

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

$4 x + 21 = - 3 \sqrt{5}$

Subtract $21$ from both sides of the equation.

$4 x = - 3 \sqrt{5} - 21$

Divide each term in $4 x = - 3 \sqrt{5} - 21$ by $4$ and simplify.

Divide each term in $4 x = - 3 \sqrt{5} - 21$ by $4$ .

$\frac{4 x}{4} = \frac{- 3 \sqrt{5}}{4} + \frac{- 21}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3 \sqrt{5}}{4} + \frac{- 21}{4}$

Divide $x$ by $1$ .

$x = \frac{- 3 \sqrt{5}}{4} + \frac{- 21}{4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{3 \sqrt{5}}{4} + \frac{- 21}{4}$

Move the negative in front of the fraction.

$x = - \frac{3 \sqrt{5}}{4} - \frac{21}{4}$

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

$x = \frac{3 \sqrt{5}}{4} - \frac{21}{4}, - \frac{3 \sqrt{5}}{4} - \frac{21}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 \sqrt{5}}{4} - \frac{21}{4}, - \frac{3 \sqrt{5}}{4} - \frac{21}{4}$

Decimal Form:

$x = - 3.57294901 \dots, - 6.92705098 \dots$

Solve Using the Square Root Property (4x+21)^2=45

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