The Solving Quadratic Equations Using Patterns Quick Check

Question

What should be the value of b in x2+bx-55=0 so that its solutions are -11 and 5?

Answer 1

To find 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 of the form \( x^2 + bx + c = 0 \):

1. The sum of the roots (solutions) is given by \( -b \).
2. The product of the roots is given by \( c \).

Given the roots \( r_1 = -11 \) and \( r_2 = 5 \):

1. Calculate the sum of the roots: \[ r_1 + r_2 = -11 + 5 = -6 \]

   According to Vieta's formula, this equals \( -b \), so: \[ -b = -6 \implies b = 6 \]

2. Calculate the product of the roots: \[ r_1 \cdot r_2 = -11 \cdot 5 = -55 \]

   This matches the constant term of the quadratic equation (which is also \(-55\)), verifying our calculations.

Thus, the value of \( b \) is \( \boxed{6} \).