Solve Using the Square Root Property (5y+2)^2+25=0

${(5 y + 2)}^{2} + 25 = 0$

Step 1

Subtract $25$ from both sides of the equation.

${(5 y + 2)}^{2} = - 25$

Step 2

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

$5 y + 2 = \pm \sqrt{- 25}$

Step 3

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

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

$5 y + 2 = \pm \sqrt{- 1 (25)}$

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

$5 y + 2 = \pm \sqrt{- 1} \cdot \sqrt{25}$

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

$5 y + 2 = \pm i \cdot \sqrt{25}$

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

$5 y + 2 = \pm i \cdot \sqrt{5^{2}}$

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

$5 y + 2 = \pm i \cdot 5$

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

$5 y + 2 = \pm 5 i$

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.

$5 y + 2 = 5 i$

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

Subtract $2$ from both sides of the equation.

$5 y = 5 i - 2$

Reorder $5 i$ and $- 2$ .

$5 y = - 2 + 5 i$

Divide each term in $5 y = - 2 + 5 i$ by $5$ and simplify.

Divide each term in $5 y = - 2 + 5 i$ by $5$ .

$\frac{5 y}{5} = \frac{- 2}{5} + \frac{5 i}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{- 2}{5} + \frac{5 i}{5}$

Divide $y$ by $1$ .

$y = \frac{- 2}{5} + \frac{5 i}{5}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{2}{5} + \frac{5 i}{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$y = - \frac{2}{5} + \frac{5 i}{5}$

Divide $i$ by $1$ .

$y = - \frac{2}{5} + i$

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

$5 y + 2 = - 5 i$

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

Subtract $2$ from both sides of the equation.

$5 y = - 5 i - 2$

Reorder $- 5 i$ and $- 2$ .

$5 y = - 2 - 5 i$

Divide each term in $5 y = - 2 - 5 i$ by $5$ and simplify.

Divide each term in $5 y = - 2 - 5 i$ by $5$ .

$\frac{5 y}{5} = \frac{- 2}{5} + \frac{- 5 i}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{- 2}{5} + \frac{- 5 i}{5}$

Divide $y$ by $1$ .

$y = \frac{- 2}{5} + \frac{- 5 i}{5}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{2}{5} + \frac{- 5 i}{5}$

Cancel the common factor of $- 5$ and $5$ .

Factor $5$ out of $- 5 i$ .

$y = - \frac{2}{5} + \frac{5 (- i)}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$y = - \frac{2}{5} + \frac{5 (- i)}{5 (1)}$

Cancel the common factor.

$y = - \frac{2}{5} + \frac{5 (- i)}{5 \cdot 1}$

Rewrite the expression.

$y = - \frac{2}{5} + \frac{- i}{1}$

Divide $- i$ by $1$ .

$y = - \frac{2}{5} - i$

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

$y = - \frac{2}{5} + i, - \frac{2}{5} - i$

Solve Using the Square Root Property (5y+2)^2+25=0

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