To solve the equation \(\sqrt{(b^2 - 10)^2} = 15\), we start by noting that the square root of a squared quantity can have two possibilities due to the properties of absolute values:
\[ |b^2 - 10| = 15 \]
This gives us two equations to solve:
- \(b^2 - 10 = 15\)
- \(b^2 - 10 = -15\)
Let's solve them one by one.
For the first equation:
\[ b^2 - 10 = 15 \] Adding 10 to both sides: \[ b^2 = 25 \] Taking the square root of both sides: \[ b = 5 \quad \text{or} \quad b = -5 \]
For the second equation:
\[ b^2 - 10 = -15 \] Adding 10 to both sides: \[ b^2 = -5 \]
Since the square of a real number cannot be negative, there are no real solutions from this equation.
Final Solutions:
Thus, the only real solutions are: \[ b = 5 \quad \text{or} \quad b = -5 \]