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 \( b \):
- Using \( 8 + 2 \):
\[ b = \frac{10}{30} = \frac{1}{3} \]
- Using \( 8 - 2 \):
\[ b = \frac{6}{30} = \frac{1}{5} \]
Thus, the solutions to the equation \( 15b^2 - 8b - 3 = -4 \) are:
\[ b = \frac{1}{3} \quad \text{and} \quad b = \frac{1}{5} \]