Question

solve by using the quadratic formula.
$y^{2} = -2y - 20$
separate your answers with commas, if necessa
express the solution set in exact simplest form.
the solution set is { } .

Explanation:

Step1: Rewrite the equation in standard form

First, we rewrite the given equation \( y^{2} = -2y - 20 \) in the standard quadratic form \( ay^{2}+by + c = 0 \).
By adding \( 2y \) and \( 20 \) to both sides, we get:
\( y^{2}+2y + 20=0 \)
Here, \( a = 1 \), \( b = 2 \), and \( c = 20 \).

Step2: Apply the quadratic formula

The quadratic formula is given by \( y=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \).
Substitute \( a = 1 \), \( b = 2 \), and \( c = 20 \) into the formula:
First, calculate the discriminant \( D=b^{2}-4ac \)
\( D=(2)^{2}-4\times1\times20 \)
\( D = 4 - 80=- 76 \)
Now, substitute into the quadratic formula:
\( y=\frac{-2\pm\sqrt{-76}}{2\times1} \)
We know that \( \sqrt{-76}=\sqrt{4\times(-19)} = 2\sqrt{-19}=2i\sqrt{19} \) (since \( \sqrt{-1}=i \))
So, \( y=\frac{-2\pm2i\sqrt{19}}{2} \)
Simplify the fraction by dividing numerator and denominator by 2:
\( y=- 1\pm i\sqrt{19} \)

Answer:

\(-1 + i\sqrt{19}, -1 - i\sqrt{19}\)