Solve Using the Quadratic Formula 9-4y^2=57

$9 - 4 y^{2} = 57$

Step 1

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

Subtract $57$ from both sides of the equation.

$9 - 4 y^{2} - 57 = 0$

Subtract $57$ from $9$ .

$- 4 y^{2} - 48 = 0$

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 $a = - 4$ , $b = 0$ , and $c = - 48$ into the quadratic formula and solve for $y$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (- 4 \cdot - 48)}}{2 \cdot - 4}$

Step 4

Simplify.

Simplify the numerator.

Raising $0$ to any positive power yields $0$ .

$y = \frac{0 \pm \sqrt{0 - 4 \cdot - 4 \cdot - 48}}{2 \cdot - 4}$

Multiply $- 4 \cdot - 4 \cdot - 48$ .

Multiply $- 4$ by $- 4$ .

$y = \frac{0 \pm \sqrt{0 + 16 \cdot - 48}}{2 \cdot - 4}$

Multiply $16$ by $- 48$ .

$y = \frac{0 \pm \sqrt{0 - 768}}{2 \cdot - 4}$

Subtract $768$ from $0$ .

$y = \frac{0 \pm \sqrt{- 768}}{2 \cdot - 4}$

Rewrite $- 768$ as $- 1 (768)$ .

$y = \frac{0 \pm \sqrt{- 1 \cdot 768}}{2 \cdot - 4}$

Rewrite $\sqrt{- 1 (768)}$ as $\sqrt{- 1} \cdot \sqrt{768}$ .

$y = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{768}}{2 \cdot - 4}$

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

$y = \frac{0 \pm i \cdot \sqrt{768}}{2 \cdot - 4}$

Rewrite $768$ as $16^{2} \cdot 3$ .

Factor $256$ out of $768$ .

$y = \frac{0 \pm i \cdot \sqrt{256 (3)}}{2 \cdot - 4}$

Rewrite $256$ as $16^{2}$ .

$y = \frac{0 \pm i \cdot \sqrt{16^{2} \cdot 3}}{2 \cdot - 4}$

Pull terms out from under the radical.

$y = \frac{0 \pm i \cdot (16 \sqrt{3})}{2 \cdot - 4}$

Move $16$ to the left of $i$ .

$y = \frac{0 \pm 16 i \sqrt{3}}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$y = \frac{0 \pm 16 i \sqrt{3}}{- 8}$

Simplify $\frac{0 \pm 16 i \sqrt{3}}{- 8}$ .

$y = \pm 2 i \sqrt{3}$