Bethany needs to solve the equation -x^2 + 4x = x - 4.
What system of equations could she write to help her solve for x?
Name the solution(s) for her system.

Correct  x₁ = - 3/2 + i √7/2  or  x₂ = - 3/2 - i √7/2

Step-by-step explanation:

We will first transform the given equation:

x² + 4x = x - 4  ⇒  x² + 4x - x + 4 = 0 ⇒ x² + 3x + 4 = 0

This equation has no solutions in the set of real numbers but it has in the set of complex numbers.

We will solve this equation as follows:

x² + 3x + 4 = x² + 2 · x · 3/2 + (3/2)² - (3/2)² + 4

the first three terms formed the square of the binomial

(x + 3/2)² - 9/4 + 4 = (x + 3/2)² - 9/4 + 16/4 = (x + 3/2)² + 7/4 =

= (x + 3/2)² -  ( - 7/4) = (x + 3/2)² -  (i √7/2)²  

we gradually transformed the given equation and get the square difference

(x + 3/2)² -  (i √7/2)² = (x + 3/2 - i √7/2) · (x + 3/2 - i √7/2)

(x + 3/2 - i √7/2) · (x + 3/2 - i √7/2) = 0 ⇒

x + 3/2 - i √7/2 = 0  or x + 3/2 + i √7/2 = 0 ⇒

x₁ = - 3/2 + i √7/2  or  x₂ = - 3/2 - i √7/2

God is with you!!!

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step-by-step explanation: