Solve Using the Square Root Property (2y+3)^2=-49

${(2 y + 3)}^{2} = - 49$

Step 1

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

$2 y + 3 = \pm \sqrt{- 49}$

Step 2

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

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

$2 y + 3 = \pm \sqrt{- 1 (49)}$

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

$2 y + 3 = \pm \sqrt{- 1} \cdot \sqrt{49}$

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

$2 y + 3 = \pm i \cdot \sqrt{49}$

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

$2 y + 3 = \pm i \cdot \sqrt{7^{2}}$

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

$2 y + 3 = \pm i \cdot 7$

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

$2 y + 3 = \pm 7 i$

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.

$2 y + 3 = 7 i$

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

Subtract $3$ from both sides of the equation.

$2 y = 7 i - 3$

Reorder $7 i$ and $- 3$ .

$2 y = - 3 + 7 i$

Divide each term in $2 y = - 3 + 7 i$ by $2$ and simplify.

Divide each term in $2 y = - 3 + 7 i$ by $2$ .

$\frac{2 y}{2} = \frac{- 3}{2} + \frac{7 i}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 3}{2} + \frac{7 i}{2}$

Divide $y$ by $1$ .

$y = \frac{- 3}{2} + \frac{7 i}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{3}{2} + \frac{7 i}{2}$

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

$2 y + 3 = - 7 i$

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

Subtract $3$ from both sides of the equation.

$2 y = - 7 i - 3$

Reorder $- 7 i$ and $- 3$ .

$2 y = - 3 - 7 i$

Divide each term in $2 y = - 3 - 7 i$ by $2$ and simplify.

Divide each term in $2 y = - 3 - 7 i$ by $2$ .

$\frac{2 y}{2} = \frac{- 3}{2} + \frac{- 7 i}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 3}{2} + \frac{- 7 i}{2}$

Divide $y$ by $1$ .

$y = \frac{- 3}{2} + \frac{- 7 i}{2}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{3}{2} + \frac{- 7 i}{2}$

Move the negative in front of the fraction.

$y = - \frac{3}{2} - \frac{7 i}{2}$

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

$y = - \frac{3}{2} + \frac{7 i}{2}, - \frac{3}{2} - \frac{7 i}{2}$

Solve Using the Square Root Property (2y+3)^2=-49

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