Solve Using the Square Root Property (7x-6)^2=-25

${(7 x - 6)}^{2} = - 25$

Step 1

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

$7 x - 6 = \pm \sqrt{- 25}$

Step 2

Simplify $\pm \sqrt{- 25}$ .

Rewrite $- 25$ as $- 1 (25)$ .

$7 x - 6 = \pm \sqrt{- 1 (25)}$

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$7 x - 6 = \pm \sqrt{- 1} \cdot \sqrt{25}$

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

$7 x - 6 = \pm i \cdot \sqrt{25}$

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

$7 x - 6 = \pm i \cdot \sqrt{5^{2}}$

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

$7 x - 6 = \pm i \cdot 5$

Move $5$ to the left of $i$ .

$7 x - 6 = \pm 5 i$

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.

$7 x - 6 = 5 i$

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

Add $6$ to both sides of the equation.

$7 x = 5 i + 6$

Reorder $5 i$ and $6$ .

$7 x = 6 + 5 i$

Divide each term in $7 x = 6 + 5 i$ by $7$ and simplify.

Divide each term in $7 x = 6 + 5 i$ by $7$ .

$\frac{7 x}{7} = \frac{6}{7} + \frac{5 i}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{6}{7} + \frac{5 i}{7}$

Divide $x$ by $1$ .

$x = \frac{6}{7} + \frac{5 i}{7}$

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

$7 x - 6 = - 5 i$

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

Add $6$ to both sides of the equation.

$7 x = - 5 i + 6$

Reorder $- 5 i$ and $6$ .

$7 x = 6 - 5 i$

Divide each term in $7 x = 6 - 5 i$ by $7$ and simplify.

Divide each term in $7 x = 6 - 5 i$ by $7$ .

$\frac{7 x}{7} = \frac{6}{7} + \frac{- 5 i}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{6}{7} + \frac{- 5 i}{7}$

Divide $x$ by $1$ .

$x = \frac{6}{7} + \frac{- 5 i}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{6}{7} - \frac{5 i}{7}$

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

$x = \frac{6}{7} + \frac{5 i}{7}, \frac{6}{7} - \frac{5 i}{7}$

Solve Using the Square Root Property (7x-6)^2=-25

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