To find the value of \( b \) in the quadratic equation \( x^2 + bx - 55 = 0 \) given that its solutions are \( -11 \) and \( 5 \), we can use Vieta's formulas. Vieta's formulas state that for a quadratic equation \( ax^2 + bx + c = 0 \), the sum and product of the roots can be used to find \( b \) and \( c \):
- The sum of the roots \( r_1 + r_2 \) is equal to \( -\frac{b}{a} \).
- The product of the roots \( r_1 \cdot r_2 \) is equal to \( \frac{c}{a} \).
In our case, \( a = 1 \) and \( c = -55 \). The roots \( r_1 \) and \( r_2 \) are \( -11 \) and \( 5 \).
Step 1: Calculate the sum of the roots
\[ r_1 + r_2 = -11 + 5 = -6 \]
According to Vieta's formulas: \[ -\frac{b}{1} = -6 \implies b = 6 \]
Step 2: Verify using the product of the roots
\[ r_1 \cdot r_2 = -11 \cdot 5 = -55 \] This matches \( c = -55 \).
Thus, the value of \( b \) is \( 6 \).
Final Answer: 6