Each solution to x^2 + 5x + 18 = 0 can be written in the form a + bi where a and b are real numbers. What is a^2 + b^2?

Guest Feb 16, 2022

#1**0 **

Instead of doing the entire problem by getting each root exactly, why not try and use Vieta's formula to get ab and a + b instead?

ab = 18 and a + b = -5. Thus, a^2 + b^2 = (a + b)^2 - 2ab. Then a^2 + b^2 = 25 - 36.

Thus, a^2 + b^2 = -11.

proyaop Feb 16, 2022

#4**+6 **

Oh yes! Thank you for your input everyone! But I caught a quick error! The solution ITSELF is a + bi, the solutions AREN'T a, and b. Thus, we might have to use the quadratic formula to find the solutions.

Applying the quadratic formula \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\). We get -5 +- sqrt(25 - 72) / 2.

Thus the two solutions are \(-{5\over2}+{\sqrt{-47}\over2}\) and \(-{5\over2} - {\sqrt{-47}\over2}\).

In this case, it doesn't matter whether b is negative or positive since we are squaring it eventually. a is 5/2. Thus, a^2 = 25/4 and b^2 = -47 or -188/4.

That means,** \(a^2 + b^2 = -{163\over4}\), **which approximates to** -40.75.**

proyaop
Feb 16, 2022