Solve Using the Square Root Property 4(x-2)^2=25

$4 {(x - 2)}^{2} = 25$

Step 1

Divide each term in $4 {(x - 2)}^{2} = 25$ by $4$ and simplify.

Divide each term in $4 {(x - 2)}^{2} = 25$ by $4$ .

$\frac{4 {(x - 2)}^{2}}{4} = \frac{25}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 {(x - 2)}^{2}}{4} = \frac{25}{4}$

Divide ${(x - 2)}^{2}$ by $1$ .

${(x - 2)}^{2} = \frac{25}{4}$

Step 2

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

$x - 2 = \pm \sqrt{\frac{25}{4}}$

Step 3

Simplify $\pm \sqrt{\frac{25}{4}}$ .

Rewrite $\sqrt{\frac{25}{4}}$ as $\frac{\sqrt{25}}{\sqrt{4}}$ .

$x - 2 = \pm \frac{\sqrt{25}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $25$ as $5^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

Simplify the denominator.

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

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

Pull terms out from under the radical, assuming positive real numbers.

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

Step 4

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.

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

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

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

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{5}{2} + 2 \cdot \frac{2}{2}$

Combine $2$ and $\frac{2}{2}$ .

$x = \frac{5}{2} + \frac{2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$x = \frac{5 + 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

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

Add $5$ and $4$ .

$x = \frac{9}{2}$

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

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

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

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

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $2$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

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

Add $- 5$ and $4$ .

$x = \frac{- 1}{2}$

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

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

$x = \frac{9}{2}, - \frac{1}{2}$

Solve Using the Square Root Property 4(x-2)^2=25

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