Question

What is the solution of the equation (x – 5)^2 + 3(x – 5) + 9 = 0? Use u substitution and the quadratic formula to solve.

A) x=-3+3i√3/2 B) x=7+3i√3/2 C) x=2 D) x=8

Answer 1

Answer:

$x = \dfrac{7 \pm 3i\sqrt{3}}{2}$

Step-by-step explanation:

(x – 5)^2 + 3(x – 5) + 9 = 0

This is a quadratic equation in x - 5.

Let u = x - 5, then the quadratic equation becomes:

u^2 + 3u + 9 = 0

We can use the quadratic formula to solve for u.

$u = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$u = \dfrac{-3 \pm \sqrt{3^2 - 4(1)(9)}}{2(1)}$

$u = \dfrac{-3 \pm \sqrt{9 - 36}}{2}$

$u = \dfrac{-3 \pm \sqrt{-27}}{2}$

$u = \dfrac{-3 \pm 3i\sqrt{3}}{2}$

Since u = x - 5, now we substitute x - 5 for u and solve for x.

$x - 5 = \dfrac{-3 \pm 3i\sqrt{3}}{2}$

$x = \dfrac{-3 \pm 3i\sqrt{3}}{2} + 5$

$x = \dfrac{-3 \pm 3i\sqrt{3}}{2} + \dfrac{10}{2}$

$x = \dfrac{7 \pm 3i\sqrt{3}}{2}$