Solve Using the Square Root Property (7y-8)^2+49=0

${(7 y - 8)}^{2} + 49 = 0$

Step 1

Subtract $49$ from both sides of the equation.

${(7 y - 8)}^{2} = - 49$

Step 2

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

$7 y - 8 = \pm \sqrt{- 49}$

Step 3

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

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

$7 y - 8 = \pm \sqrt{- 1 (49)}$

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

$7 y - 8 = \pm \sqrt{- 1} \cdot \sqrt{49}$

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

$7 y - 8 = \pm i \cdot \sqrt{49}$

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

$7 y - 8 = \pm i \cdot \sqrt{7^{2}}$

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

$7 y - 8 = \pm i \cdot 7$

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

$7 y - 8 = \pm 7 i$

Step 4

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.

$7 y - 8 = 7 i$

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

Add $8$ to both sides of the equation.

$7 y = 7 i + 8$

Reorder $7 i$ and $8$ .

$7 y = 8 + 7 i$

Divide each term in $7 y = 8 + 7 i$ by $7$ and simplify.

Divide each term in $7 y = 8 + 7 i$ by $7$ .

$\frac{7 y}{7} = \frac{8}{7} + \frac{7 i}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{8}{7} + \frac{7 i}{7}$

Divide $y$ by $1$ .

$y = \frac{8}{7} + \frac{7 i}{7}$

Simplify the right side.

Cancel the common factor of $7$ .

Cancel the common factor.

$y = \frac{8}{7} + \frac{7 i}{7}$

Divide $i$ by $1$ .

$y = \frac{8}{7} + i$

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

$7 y - 8 = - 7 i$

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

Add $8$ to both sides of the equation.

$7 y = - 7 i + 8$

Reorder $- 7 i$ and $8$ .

$7 y = 8 - 7 i$

Divide each term in $7 y = 8 - 7 i$ by $7$ and simplify.

Divide each term in $7 y = 8 - 7 i$ by $7$ .

$\frac{7 y}{7} = \frac{8}{7} + \frac{- 7 i}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{8}{7} + \frac{- 7 i}{7}$

Divide $y$ by $1$ .

$y = \frac{8}{7} + \frac{- 7 i}{7}$

Simplify the right side.

Cancel the common factor of $- 7$ and $7$ .

Factor $7$ out of $- 7 i$ .

$y = \frac{8}{7} + \frac{7 (- i)}{7}$

Cancel the common factors.

Factor $7$ out of $7$ .

$y = \frac{8}{7} + \frac{7 (- i)}{7 (1)}$

Cancel the common factor.

$y = \frac{8}{7} + \frac{7 (- i)}{7 \cdot 1}$

Rewrite the expression.

$y = \frac{8}{7} + \frac{- i}{1}$

Divide $- i$ by $1$ .

$y = \frac{8}{7} - i$

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

$y = \frac{8}{7} + i, \frac{8}{7} - i$

Solve Using the Square Root Property (7y-8)^2+49=0

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