Bot GPT 4-o mini To find the value of \( b \) in the quadratic equation \( x^2 + bx - 55 = 0 \) such that its solutions are \( -11 \) and \( 5 \), we can use the fact that if \( p \) and \( q \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \), then:
- The sum of the roots \( p + q = -\frac{b}{a} \).
- The product of the roots \( pq = \frac{c}{a} \).
In our case, the specific values are:
- \( p = -11 \)
- \( q = 5 \)
- \( a = 1 \)
- \( c = -55 \)
First, calculate the sum of the roots: \[ p + q = -11 + 5 = -6 \] According to the formula for the sum of the roots, we have: \[ -6 = -b \quad \Rightarrow \quad b = 6 \]
Next, calculate the product of the roots: \[ pq = (-11)(5) = -55 \] According to the formula for the product of the roots, we have: \[ -55 = -55 \] This confirms the values we derived.
Thus, the value of \( b \) is \( 6 \).
So the answer is \( \boxed{6} \).
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