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

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

Step 1

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

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

Step 2

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

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

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

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

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

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

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

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

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

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

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

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

$6 x - 7 = \pm 5 i$

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.

$6 x - 7 = 5 i$

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

Add $7$ to both sides of the equation.

$6 x = 5 i + 7$

Reorder $5 i$ and $7$ .

$6 x = 7 + 5 i$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

$6 x - 7 = - 5 i$

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

Add $7$ to both sides of the equation.

$6 x = - 5 i + 7$

Reorder $- 5 i$ and $7$ .

$6 x = 7 - 5 i$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

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

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