Question

question
use the quadratic formula to solve. express your answer in simplest form.
$-b^2 - 18b - 8 = -6b^2$

Explanation:

Step1: Rearrange the equation

First, we need to rewrite the given equation \(-b^{2}-18b - 8=-6b^{2}\) in the standard quadratic form \(ax^{2}+bx + c = 0\) (here we use \(b\) as the variable, so we can think of it as \(ab^{2}+bb + c=0\), but to avoid confusion, we will use \(x\) - like notation for the formula, but the variable is \(b\)).

Add \(6b^{2}\) to both sides of the equation:
\(-b^{2}+6b^{2}-18b - 8=0\)
Combine like terms: \(5b^{2}-18b - 8 = 0\)

Now we have a quadratic equation in the form \(ax^{2}+bx + c = 0\) where \(a = 5\), \(b=- 18\) (note that here the coefficient of \(b\) is \(-18\), and the variable is \(b\), so we have to be careful with the notation), and \(c=-8\).

The quadratic formula is \(x=\frac{-B\pm\sqrt{B^{2}-4AC}}{2A}\) (where for our equation \(A = 5\), \(B=-18\), \(C = - 8\))

Step2: Identify \(a\), \(b\), \(c\)

For the quadratic equation \(5b^{2}-18b - 8=0\), we have:
\(a = 5\), \(b=-18\), \(c=-8\)

Step3: Apply the quadratic formula

The quadratic formula is \(b=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) (now using \(b\) as the variable, so the formula is \(b=\frac{-B\pm\sqrt{B^{2}-4AC}}{2A}\) where \(A = 5\), \(B=-18\), \(C=-8\))

First, calculate the discriminant \(D=B^{2}-4AC\)

Substitute \(A = 5\), \(B=-18\), \(C = - 8\) into the discriminant formula:

\(D=(-18)^{2}-4\times5\times(-8)\)
\(=324 + 160\)
\(=484\)

Then, find the square root of the discriminant: \(\sqrt{484}=22\)

Now, substitute into the quadratic formula:

\(b=\frac{-(-18)\pm22}{2\times5}=\frac{18\pm22}{10}\)

Step4: Find the two solutions

We have two cases:

Case 1: \(b=\frac{18 + 22}{10}=\frac{40}{10}=4\)

Case 2: \(b=\frac{18-22}{10}=\frac{-4}{10}=-\frac{2}{5}\)

Answer:

The solutions are \(b = 4\) and \(b=-\frac{2}{5}\)