Solve Using the Square Root Property (4x+1)^2=19

${(4 x + 1)}^{2} = 19$

Step 1

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

$4 x + 1 = \pm \sqrt{19}$

Step 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.

$4 x + 1 = \sqrt{19}$

Subtract $1$ from both sides of the equation.

$4 x = \sqrt{19} - 1$

Divide each term in $4 x = \sqrt{19} - 1$ by $4$ and simplify.

Divide each term in $4 x = \sqrt{19} - 1$ by $4$ .

$\frac{4 x}{4} = \frac{\sqrt{19}}{4} + \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{\sqrt{19}}{4} + \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{\sqrt{19}}{4} + \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{\sqrt{19}}{4} - \frac{1}{4}$

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

$4 x + 1 = - \sqrt{19}$

Subtract $1$ from both sides of the equation.

$4 x = - \sqrt{19} - 1$

Divide each term in $4 x = - \sqrt{19} - 1$ by $4$ and simplify.

Divide each term in $4 x = - \sqrt{19} - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- \sqrt{19}}{4} + \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- \sqrt{19}}{4} + \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{- \sqrt{19}}{4} + \frac{- 1}{4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{\sqrt{19}}{4} + \frac{- 1}{4}$

Move the negative in front of the fraction.

$x = - \frac{\sqrt{19}}{4} - \frac{1}{4}$

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

$x = \frac{\sqrt{19}}{4} - \frac{1}{4}, - \frac{\sqrt{19}}{4} - \frac{1}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{19}}{4} - \frac{1}{4}, - \frac{\sqrt{19}}{4} - \frac{1}{4}$

Decimal Form:

$x = 0.83972473 \dots, - 1.33972473 \dots$

Solve Using the Square Root Property (4x+1)^2=19

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