Question

What should be the value of b in x2+bx−55=0
so that its solutions are −11
and 5? (1 point)
Responses

−6
negative 6

6
6

16
16

−16

Answer 1

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:

1. \( x_1 + x_2 = -\frac{b}{a} \)
2. \( x_1 \cdot x_2 = \frac{c}{a} \)

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:

\[
\boxed{6}
\]