To find the value of \( b \) in the equation \( x^2 + bx - 55 = 0 \) such that its solutions are \( -11 \) and \( 5 \), we can use Vieta's formulas. According to Vieta's formulas, for a quadratic equation of the form \( x^2 + px + q = 0 \), the sum of the roots \( r_1 \) and \( r_2 \) is given by:
\[ r_1 + r_2 = -p \]
and the product of the roots \( r_1 \) and \( r_2 \) is given by:
\[ r_1 \cdot r_2 = q. \]
In our case, the roots are \( -11 \) and \( 5 \).
- Calculate the sum of the roots:
\[ r_1 + r_2 = -11 + 5 = -6. \]
According to Vieta's, this sum equals \( -b \):
\[ -b = -6 \implies b = 6. \]
- Verify the product of the roots:
We can also verify the product of the roots:
\[ r_1 \cdot r_2 = -11 \cdot 5 = -55. \]
According to Vieta's, this product equals \( -55 \) as indicated by the coefficient in the equation.
Given the calculations and verification, the value of \( b \) is:
\[ \boxed{6}. \]