How do you find the roots, real and imaginary, of y= x^2 - 5x - (2x-2)^2 using the quadratic formula?

1 Answer
Alan P.
Feb 17, 2016

There are two imaginary roots:
color(white)("XXX")1/2+sqrt(39)/6i and 1/2-sqrt(39)/6i

Explanation:

First expand the expression given on the left into standard form:
y=x^2-5x-(2x-2)^2
color(white)("XXX")y=x^2-5x-(4x^2-8x+4)

color(white)("XXX")y=-3x^2+3x-4

This is in standard form: y=ax^2+bx+c for which the quadratic formula tells us the roots are:
color(white)("XXX")x=(-b+-sqrt(b^2-4ac))/(2a)

For this specific example, the roots are
color(white)("XXX")x=(-3+-sqrt(3^2-4(-3)(-4)))/(2(-3)

color(white)("XXXX")=(-3+-sqrt(-39))/(-6)

color(white)("XXXX")=1/2+-sqrt(39)/6i

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