Question

Use the quadratic formula to solve. Express your answer in simplest form.
15, b, squared, minus, 8, b, minus, 3, equals, minus, 4
15b
2
−8b−3=
−4

Answer 1

To solve the equation \( 15b^2 - 8b - 3 = -4 \), we first need to rearrange the equation to set it to zero. We can do this by adding 4 to both sides:

\[
15b^2 - 8b - 3 + 4 = 0
\]

This simplifies to:

\[
15b^2 - 8b + 1 = 0
\]

Now, we will use the quadratic formula, which is:

\[
b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\]

In our equation, \( A = 15 \), \( B = -8 \), and \( C = 1 \). We can now calculate \( B^2 - 4AC \):

\[
B^2 = (-8)^2 = 64
\]
\[
4AC = 4 \cdot 15 \cdot 1 = 60
\]
\[
B^2 - 4AC = 64 - 60 = 4
\]

Now we can substitute \( A \), \( B \), and \( B^2 - 4AC \) into the quadratic formula:

\[
b = \frac{-(-8) \pm \sqrt{4}}{2 \cdot 15}
\]

This simplifies to:

\[
b = \frac{8 \pm 2}{30}
\]

Now we solve for the two possible values of \( b \):

1. Using \( 8 + 2 \):

\[
b = \frac{10}{30} = \frac{1}{3}
\]

2. Using \( 8 - 2 \):

\[
b = \frac{6}{30} = \frac{1}{5}
\]

Thus, the solutions to the equation \( 15b^2 - 8b - 3 = -4 \) are:

\[
b = \frac{1}{3} \quad \text{and} \quad b = \frac{1}{5}
\]