Solve Using the Square Root Property (4x+1)^2=20

${(4 x + 1)}^{2} = 20$

Step 1

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

$4 x + 1 = \pm \sqrt{20}$

Step 2

Simplify $\pm \sqrt{20}$ .

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$4 x + 1 = \pm \sqrt{4 (5)}$

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

$4 x + 1 = \pm \sqrt{2^{2} \cdot 5}$

Pull terms out from under the radical.

$4 x + 1 = \pm 2 \sqrt{5}$

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.

$4 x + 1 = 2 \sqrt{5}$

Subtract $1$ from both sides of the equation.

$4 x = 2 \sqrt{5} - 1$

Divide each term in $4 x = 2 \sqrt{5} - 1$ by $4$ and simplify.

Divide each term in $4 x = 2 \sqrt{5} - 1$ by $4$ .

$\frac{4 x}{4} = \frac{2 \sqrt{5}}{4} + \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{2 \sqrt{5}}{4} + \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{2 \sqrt{5}}{4} + \frac{- 1}{4}$

Simplify the right side.

Simplify each term.

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

Factor $2$ out of $2 \sqrt{5}$ .

$x = \frac{2 \sqrt{5}}{4} + \frac{- 1}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = \frac{2 \sqrt{5}}{2 \cdot 2} + \frac{- 1}{4}$

Cancel the common factor.

$x = \frac{2 \sqrt{5}}{2 \cdot 2} + \frac{- 1}{4}$

Rewrite the expression.

$x = \frac{\sqrt{5}}{2} + \frac{- 1}{4}$

Move the negative in front of the fraction.

$x = \frac{\sqrt{5}}{2} - \frac{1}{4}$

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

$4 x + 1 = - 2 \sqrt{5}$

Subtract $1$ from both sides of the equation.

$4 x = - 2 \sqrt{5} - 1$

Divide each term in $4 x = - 2 \sqrt{5} - 1$ by $4$ and simplify.

Divide each term in $4 x = - 2 \sqrt{5} - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 2 \sqrt{5}}{4} + \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 2 \sqrt{5}}{4} + \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{- 2 \sqrt{5}}{4} + \frac{- 1}{4}$

Simplify the right side.

Simplify each term.

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

Factor $2$ out of $- 2 \sqrt{5}$ .

$x = \frac{2 (- \sqrt{5})}{4} + \frac{- 1}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = \frac{2 (- \sqrt{5})}{2 (2)} + \frac{- 1}{4}$

Cancel the common factor.

$x = \frac{2 (- \sqrt{5})}{2 \cdot 2} + \frac{- 1}{4}$

Rewrite the expression.

$x = \frac{- \sqrt{5}}{2} + \frac{- 1}{4}$

Move the negative in front of the fraction.

$x = - \frac{\sqrt{5}}{2} + \frac{- 1}{4}$

Move the negative in front of the fraction.

$x = - \frac{\sqrt{5}}{2} - \frac{1}{4}$

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

$x = \frac{\sqrt{5}}{2} - \frac{1}{4}, - \frac{\sqrt{5}}{2} - \frac{1}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{5}}{2} - \frac{1}{4}, - \frac{\sqrt{5}}{2} - \frac{1}{4}$

Decimal Form:

$x = 0.86803398 \dots, - 1.36803398 \dots$

Solve Using the Square Root Property (4x+1)^2=20

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