Solve Using the Square Root Property y^2-6y+9=-16

$y^{2} - 6 y + 9 = - 16$

Step 1

Move all terms to the left side of the equation and simplify.

Add $16$ to both sides of the equation.

$y^{2} - 6 y + 9 + 16 = 0$

Add $9$ and $16$ .

$y^{2} - 6 y + 25 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = - 6$ , and $c = 25$ into the quadratic formula and solve for $y$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 25$ .

Multiply $- 4$ by $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 25}}{2 \cdot 1}$

Multiply $- 4$ by $25$ .

$y = \frac{6 \pm \sqrt{36 - 100}}{2 \cdot 1}$

Subtract $100$ from $36$ .

$y = \frac{6 \pm \sqrt{- 64}}{2 \cdot 1}$

Rewrite $- 64$ as $- 1 (64)$ .

$y = \frac{6 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$y = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 1}$

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

$y = \frac{6 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$y = \frac{6 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

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

$y = \frac{6 \pm i \cdot 8}{2 \cdot 1}$

Move $8$ to the left of $i$ .

$y = \frac{6 \pm 8 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{6 \pm 8 i}{2}$

Simplify $\frac{6 \pm 8 i}{2}$ .

$y = 3 \pm 4 i$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 25$ .

Multiply $- 4$ by $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 25}}{2 \cdot 1}$

Multiply $- 4$ by $25$ .

$y = \frac{6 \pm \sqrt{36 - 100}}{2 \cdot 1}$

Subtract $100$ from $36$ .

$y = \frac{6 \pm \sqrt{- 64}}{2 \cdot 1}$

Rewrite $- 64$ as $- 1 (64)$ .

$y = \frac{6 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$y = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 1}$

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

$y = \frac{6 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$y = \frac{6 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

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

$y = \frac{6 \pm i \cdot 8}{2 \cdot 1}$

Move $8$ to the left of $i$ .

$y = \frac{6 \pm 8 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{6 \pm 8 i}{2}$

Simplify $\frac{6 \pm 8 i}{2}$ .

$y = 3 \pm 4 i$

Change the $\pm$ to $+$ .

$y = 3 + 4 i$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 25$ .

Multiply $- 4$ by $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 25}}{2 \cdot 1}$

Multiply $- 4$ by $25$ .

$y = \frac{6 \pm \sqrt{36 - 100}}{2 \cdot 1}$

Subtract $100$ from $36$ .

$y = \frac{6 \pm \sqrt{- 64}}{2 \cdot 1}$

Rewrite $- 64$ as $- 1 (64)$ .

$y = \frac{6 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$y = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 1}$

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

$y = \frac{6 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$y = \frac{6 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

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

$y = \frac{6 \pm i \cdot 8}{2 \cdot 1}$

Move $8$ to the left of $i$ .

$y = \frac{6 \pm 8 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{6 \pm 8 i}{2}$

Simplify $\frac{6 \pm 8 i}{2}$ .

$y = 3 \pm 4 i$

Change the $\pm$ to $-$ .

$y = 3 - 4 i$

Step 7

The final answer is the combination of both solutions.

$y = 3 + 4 i, 3 - 4 i$

Solve Using the Square Root Property y^2-6y+9=-16

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