To find the value of \( b \) in the quadratic equation \( x^2 + bx - 55 = 0 \) given that the solutions are \( -11 \) and \( 5 \), we can use Vieta's formulas.
According to Vieta's formulas, for a quadratic equation of the form \( x^2 + bx + c = 0 \):
- The sum of the roots \( r_1 + r_2 = -b \)
- The product of the roots \( r_1 \cdot r_2 = c \)
In our case, the roots are \( r_1 = -11 \) and \( r_2 = 5 \).
-
Calculate the sum of the roots: \[ -11 + 5 = -6 \] According to Vieta's, this gives us: \[ -b = -6 \quad \Rightarrow \quad b = 6 \]
-
Calculate the product of the roots: \[ -11 \cdot 5 = -55 \] This verifies the constant term \( c = -55 \).
Thus, the value of \( b \) that results in the roots of \( -11 \) and \( 5 \) for the equation \( x^2 + bx - 55 = 0 \) is:
\[ \boxed{6} \]