To determine the value of \( b \) in the quadratic equation \( x^2 + bx - 55 = 0 \) with solutions \( -11 \) and \( 5 \), we can use Vieta's formulas. According to Vieta's, for a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots (\( r_1 + r_2 \)) is given by \( -\frac{b}{a} \) and the product of the roots (\( r_1 \cdot r_2 \)) is given by \( \frac{c}{a} \).
Given roots:
- \( r_1 = -11 \)
- \( r_2 = 5 \)
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Find the sum of the roots: \[ r_1 + r_2 = -11 + 5 = -6 \]
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Find the product of the roots: \[ r_1 \cdot r_2 = -11 \cdot 5 = -55 \]
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For the product of the roots: From Vieta’s formula, since \( a = 1 \) and \( c = -55 \): \[ \frac{c}{a} = -55 \quad \text{(which matches)} \]
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From the sum of the roots, we have: \[ -\frac{b}{1} = -6 \quad \Rightarrow \quad -b = -6 \quad \Rightarrow \quad b = 6 \]
Thus, the value of \( b \) should be 6.