Question

solve the quadratic equation using any method. give exact values - no decimals. use a comma to separate your two answers.\\( y^2 + 10y = -20 \\)\\( y = \\)

Explanation:

Step1: Rewrite in standard form

First, rewrite the quadratic equation \( y^{2}+10y = - 20\) in standard form \(ax^{2}+bx + c=0\) (here \(x = y\)). Add 20 to both sides:
\(y^{2}+10y + 20=0\)

Step2: Use quadratic formula

The quadratic formula for a quadratic equation \(ax^{2}+bx + c = 0\) is \(y=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). For the equation \(y^{2}+10y + 20 = 0\), we have \(a = 1\), \(b=10\), and \(c = 20\).
First, calculate the discriminant \(\Delta=b^{2}-4ac\):
\(\Delta=(10)^{2}-4\times1\times20=100 - 80 = 20\)
Then, substitute \(a\), \(b\), and \(\Delta\) into the quadratic formula:
\(y=\frac{-10\pm\sqrt{20}}{2\times1}=\frac{-10\pm2\sqrt{5}}{2}=- 5\pm\sqrt{5}\)

Answer:

\(-5 + \sqrt{5},-5-\sqrt{5}\)