Question

The roots of the quadratic equation x2 + 4x + K = 0 are x 1 and x 2. If x 12 - x 22 = -32, then find K.

Answer 1

$\displaystyle x_1^2-x_2^2=-32\\(x_1-x_2)(x_1+x_2)=-32\\\left(\pm\frac{\sqrt{D}}{a}\right)\left(-\frac{b}{a}\right)=-32\\\left(\pm\frac{\sqrt{b^2-4ac}}{a}\right)\left(-\frac{b}{a}\right)=-32\\\left(\pm\frac{\sqrt{(-4)^2-4(1)k}}{1}\right)\left(-\frac{4}{1}\right)=-32\\\pm\sqrt{16-4k}=8\\16-4k=64\\-4k=48\\\boxed{\boxed{k=-12}}$