Question

solve the equation $-2s^2 - 20s = 44$. present exact values of the solutions. if there are more than one solutions, separate them by a comma.
$s = $
question help: video

Explanation:

Step1: Rewrite the equation in standard form

First, we want to rewrite the quadratic equation \(-2s^{2}-20s = 44\) in the standard form \(ax^{2}+bx + c=0\). We can do this by subtracting 44 from both sides of the equation:
\(-2s^{2}-20s - 44=0\)
We can also simplify the equation by dividing every term by - 2 to make the coefficients simpler:
\(\frac{-2s^{2}}{-2}+\frac{-20s}{-2}+\frac{-44}{-2}=0\)
Which simplifies to:
\(s^{2}+10s + 22 = 0\)

Step2: Use the quadratic formula

For a quadratic equation of the form \(ax^{2}+bx + c = 0\), the quadratic formula is given by \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). In our equation \(s^{2}+10s + 22=0\), we have \(a = 1\), \(b=10\) and \(c = 22\).
First, we calculate the discriminant \(\Delta=b^{2}-4ac\). Substituting the values of \(a\), \(b\) and \(c\) we get:
\(\Delta=(10)^{2}-4\times1\times22\)
\(=100 - 88\)
\(=12\)
Now we substitute \(a = 1\), \(b = 10\) and \(\Delta=12\) into the quadratic formula:
\(s=\frac{-10\pm\sqrt{12}}{2\times1}\)
We can simplify \(\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\), so:
\(s=\frac{-10\pm2\sqrt{3}}{2}\)
We can factor out a 2 from the numerator:
\(s=\frac{2(-5\pm\sqrt{3})}{2}\)
Then we cancel out the 2 in the numerator and the denominator:
\(s=- 5\pm\sqrt{3}\)

Answer:

\(-5 + \sqrt{3},-5-\sqrt{3}\)