Explore: The Square Root Property to Solve Solutions

The Square Root Property to Solve Solutions

The square root property can be a more efficient method to solve equations when in the form $a x^{2} + c = 0$ or $a {(x - d)}^{2} + c = 0$ . 
The squared-term or expression will be isolated and then you solve for the variable.

Example 9

Solve.

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

Implement

$\begin{array}{rcl} \sqrt{{(x + 2)}^{2}} & = & \pm \sqrt{\frac{- 2}{25}} \\ x + 2 & = & \pm \frac{i \sqrt{2}}{5} \end{array}$

Explain

Square root property

$\begin{array}{rcl} x & = & - 2 \pm \frac{i \sqrt{2}}{5} \end{array}$

Example 10

Solve using the square root property.

$4 x^{2} + 80 = 0$

Implement

$4 x^{2} = - 80 x^{2} = - 20 \sqrt{x^{2}} = \sqrt{- 20} x = \pm 2 i \sqrt{5}$

Explain

Isolate the squared expression
 Solve for x

Note

This problem can also be solved by factoring, but the directions say to solve using the square root property.

Example 11

Solve.

${(3 x + 5)}^{2} - 144 = 0$

Implement

$\begin{array}{rcl} {(3 x + 5)}^{2} & = & 144 \\ \sqrt{{(3 x + 5)}^{2}} & = & \pm \sqrt{144} \\ 3 x + 5 & = & \pm 12 \\ 3 x & = & - 5 \pm 12 \\ x & = & \frac{- 5 \pm 12}{3} \\ x & = & - \frac{17}{3}, \frac{7}{3} \end{array}$

Explain

Isolate the squared expression 
Square root property 
Solve for x