Solve Using the Square Root Property (6x-1)^2=-49

${(6 x - 1)}^{2} = - 49$

Step 1

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

$6 x - 1 = \pm \sqrt{- 49}$

Step 2

Simplify $\pm \sqrt{- 49}$ .

Rewrite $- 49$ as $- 1 (49)$ .

$6 x - 1 = \pm \sqrt{- 1 (49)}$

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$6 x - 1 = \pm \sqrt{- 1} \cdot \sqrt{49}$

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

$6 x - 1 = \pm i \cdot \sqrt{49}$

Rewrite $49$ as $7^{2}$ .

$6 x - 1 = \pm i \cdot \sqrt{7^{2}}$

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

$6 x - 1 = \pm i \cdot 7$

Move $7$ to the left of $i$ .

$6 x - 1 = \pm 7 i$

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.

$6 x - 1 = 7 i$

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

Add $1$ to both sides of the equation.

$6 x = 7 i + 1$

Reorder $7 i$ and $1$ .

$6 x = 1 + 7 i$

Divide each term in $6 x = 1 + 7 i$ by $6$ and simplify.

Divide each term in $6 x = 1 + 7 i$ by $6$ .

$\frac{6 x}{6} = \frac{1}{6} + \frac{7 i}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{1}{6} + \frac{7 i}{6}$

Divide $x$ by $1$ .

$x = \frac{1}{6} + \frac{7 i}{6}$

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

$6 x - 1 = - 7 i$

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

Add $1$ to both sides of the equation.

$6 x = - 7 i + 1$

Reorder $- 7 i$ and $1$ .

$6 x = 1 - 7 i$

Divide each term in $6 x = 1 - 7 i$ by $6$ and simplify.

Divide each term in $6 x = 1 - 7 i$ by $6$ .

$\frac{6 x}{6} = \frac{1}{6} + \frac{- 7 i}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{1}{6} + \frac{- 7 i}{6}$

Divide $x$ by $1$ .

$x = \frac{1}{6} + \frac{- 7 i}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{1}{6} - \frac{7 i}{6}$

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

$x = \frac{1}{6} + \frac{7 i}{6}, \frac{1}{6} - \frac{7 i}{6}$

Solve Using the Square Root Property (6x-1)^2=-49

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