Solve Using the Square Root Property (7y-9)^2+100=0

${(7 y - 9)}^{2} + 100 = 0$

Step 1

Subtract $100$ from both sides of the equation.

${(7 y - 9)}^{2} = - 100$

Step 2

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

$7 y - 9 = \pm \sqrt{- 100}$

Step 3

Simplify $\pm \sqrt{- 100}$ .

Rewrite $- 100$ as $- 1 (100)$ .

$7 y - 9 = \pm \sqrt{- 1 (100)}$

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$7 y - 9 = \pm \sqrt{- 1} \cdot \sqrt{100}$

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

$7 y - 9 = \pm i \cdot \sqrt{100}$

Rewrite $100$ as $10^{2}$ .

$7 y - 9 = \pm i \cdot \sqrt{10^{2}}$

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

$7 y - 9 = \pm i \cdot 10$

Move $10$ to the left of $i$ .

$7 y - 9 = \pm 10 i$

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.

$7 y - 9 = 10 i$

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

Add $9$ to both sides of the equation.

$7 y = 10 i + 9$

Reorder $10 i$ and $9$ .

$7 y = 9 + 10 i$

Divide each term in $7 y = 9 + 10 i$ by $7$ and simplify.

Divide each term in $7 y = 9 + 10 i$ by $7$ .

$\frac{7 y}{7} = \frac{9}{7} + \frac{10 i}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{9}{7} + \frac{10 i}{7}$

Divide $y$ by $1$ .

$y = \frac{9}{7} + \frac{10 i}{7}$

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

$7 y - 9 = - 10 i$

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

Add $9$ to both sides of the equation.

$7 y = - 10 i + 9$

Reorder $- 10 i$ and $9$ .

$7 y = 9 - 10 i$

Divide each term in $7 y = 9 - 10 i$ by $7$ and simplify.

Divide each term in $7 y = 9 - 10 i$ by $7$ .

$\frac{7 y}{7} = \frac{9}{7} + \frac{- 10 i}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{9}{7} + \frac{- 10 i}{7}$

Divide $y$ by $1$ .

$y = \frac{9}{7} + \frac{- 10 i}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{9}{7} - \frac{10 i}{7}$

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

$y = \frac{9}{7} + \frac{10 i}{7}, \frac{9}{7} - \frac{10 i}{7}$

Solve Using the Square Root Property (7y-9)^2+100=0

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