Solve Using the Square Root Property (2x-4)^2=23

${(2 x - 4)}^{2} = 23$

Step 1

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

$2 x - 4 = \pm \sqrt{23}$

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.

$2 x - 4 = \sqrt{23}$

Add $4$ to both sides of the equation.

$2 x = \sqrt{23} + 4$

Divide each term in $2 x = \sqrt{23} + 4$ by $2$ and simplify.

Divide each term in $2 x = \sqrt{23} + 4$ by $2$ .

$\frac{2 x}{2} = \frac{\sqrt{23}}{2} + \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\sqrt{23}}{2} + \frac{4}{2}$

Divide $x$ by $1$ .

$x = \frac{\sqrt{23}}{2} + \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x = \frac{\sqrt{23}}{2} + 2$

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

$2 x - 4 = - \sqrt{23}$

Add $4$ to both sides of the equation.

$2 x = - \sqrt{23} + 4$

Divide each term in $2 x = - \sqrt{23} + 4$ by $2$ and simplify.

Divide each term in $2 x = - \sqrt{23} + 4$ by $2$ .

$\frac{2 x}{2} = \frac{- \sqrt{23}}{2} + \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \sqrt{23}}{2} + \frac{4}{2}$

Divide $x$ by $1$ .

$x = \frac{- \sqrt{23}}{2} + \frac{4}{2}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{\sqrt{23}}{2} + \frac{4}{2}$

Divide $4$ by $2$ .

$x = - \frac{\sqrt{23}}{2} + 2$

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

$x = \frac{\sqrt{23}}{2} + 2, - \frac{\sqrt{23}}{2} + 2$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{23}}{2} + 2, - \frac{\sqrt{23}}{2} + 2$

Decimal Form:

$x = 4.39791576 \dots, - 0.39791576 \dots$

Solve Using the Square Root Property (2x-4)^2=23

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