To find the value of \( b \) in the quadratic equation \( x^2 + bx - 55 = 0 \) given the solutions \( x_1 = -11 \) and \( x_2 = 5 \), we can use Vieta's formulas. According to Vieta's, the sum and product of the roots of the equation \( ax^2 + bx + c = 0 \) can give us the relationships:
- \( x_1 + x_2 = -\frac{b}{a} \)
- \( x_1 \cdot x_2 = \frac{c}{a} \)
Here, we have:
- \( x_1 = -11 \)
- \( x_2 = 5 \)
- \( a = 1 \) (the coefficient of \( x^2 \))
- \( c = -55 \)
First, we calculate the sum of the roots:
\[ x_1 + x_2 = -11 + 5 = -6 \]
Using the relationship from Vieta’s formulas:
\[ x_1 + x_2 = -\frac{b}{1} \implies -6 = -b \implies b = 6 \]
Next, we check the product of the roots:
\[ x_1 \cdot x_2 = -11 \cdot 5 = -55 \]
Using the second relationship from Vieta's formulas:
\[ x_1 \cdot x_2 = \frac{-55}{1} \implies -55 = -55 \]
This is consistent.
Thus, the value of \( b \) should be:
\[ \boxed{6} \]
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