Solve Using the Square Root Property (2x-7)^2=-9

${(2 x - 7)}^{2} = - 9$

Step 1

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

$2 x - 7 = \pm \sqrt{- 9}$

Step 2

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$2 x - 7 = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$2 x - 7 = \pm \sqrt{- 1} \cdot \sqrt{9}$

Rewrite $\sqrt{- 1}$ as $i$ .

$2 x - 7 = \pm i \cdot \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

$2 x - 7 = \pm i \cdot \sqrt{3^{2}}$

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

$2 x - 7 = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$2 x - 7 = \pm 3 i$

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.

$2 x - 7 = 3 i$

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

Add $7$ to both sides of the equation.

$2 x = 3 i + 7$

Reorder $3 i$ and $7$ .

$2 x = 7 + 3 i$

Divide each term in $2 x = 7 + 3 i$ by $2$ and simplify.

Divide each term in $2 x = 7 + 3 i$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2} + \frac{3 i}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2} + \frac{3 i}{2}$

Divide $x$ by $1$ .

$x = \frac{7}{2} + \frac{3 i}{2}$

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

$2 x - 7 = - 3 i$

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

Add $7$ to both sides of the equation.

$2 x = - 3 i + 7$

Reorder $- 3 i$ and $7$ .

$2 x = 7 - 3 i$

Divide each term in $2 x = 7 - 3 i$ by $2$ and simplify.

Divide each term in $2 x = 7 - 3 i$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2} + \frac{- 3 i}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2} + \frac{- 3 i}{2}$

Divide $x$ by $1$ .

$x = \frac{7}{2} + \frac{- 3 i}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{7}{2} - \frac{3 i}{2}$

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

$x = \frac{7}{2} + \frac{3 i}{2}, \frac{7}{2} - \frac{3 i}{2}$

Solve Using the Square Root Property (2x-7)^2=-9

Related Questions

Support LCMGCF.com by disabling your adblocker.