Solve Using the Quadratic Formula y^2+9=-9y

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

Step 1

Add $9 y$ to both sides of the equation.

$y^{2} + 9 + 9 y = 0$

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 $a = 1$ , $b = 9$ , and $c = 9$ into the quadratic formula and solve for $y$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $9$ to the power of $2$ .

$y = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 9$ .

Multiply $- 4$ by $1$ .

$y = \frac{- 9 \pm \sqrt{81 - 4 \cdot 9}}{2 \cdot 1}$

Multiply $- 4$ by $9$ .

$y = \frac{- 9 \pm \sqrt{81 - 36}}{2 \cdot 1}$

Subtract $36$ from $81$ .

$y = \frac{- 9 \pm \sqrt{45}}{2 \cdot 1}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$y = \frac{- 9 \pm \sqrt{9 (5)}}{2 \cdot 1}$

Rewrite $9$ as $3^{2}$ .

$y = \frac{- 9 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{- 9 \pm 3 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{- 9 \pm 3 \sqrt{5}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{- 9 \pm 3 \sqrt{5}}{2}$

Decimal Form:

$y = - 1.14589803 \dots, - 7.85410196 \dots$