Question

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

Answer 1

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`