Question

How do you solve `(-2x+5)^2 = -8`?

Answer 1

See a solution process below:

Explanation:

First, expand the term on the left using this special rule for multiplying quadratics:

`(color(red)(x) + color(blue)(y))^2 = color(red)(x)^2 + 2color(red)(x)color(blue)(y) + color(blue)(y)^2`

Substituting gives:

`(color(red)(-2x) + color(blue)(5))^2 = -8`

`(color(red)(-2x))^2 + (2 * color(red)(-2x) * color(blue)(5)) + color(blue)(5)^2 = -8`

`4x^2 + (-20x) + 25 = -8`

`4x^2 - 20x + 25 = -8`

We can next convert this to standard form:

`4x^2 - 20x + 25 + color(red)(8) = -8 + color(red)(8)`

`4x^2 - 20x + 33 = 0`

We can now use the quadratic formula to find the solutions for `x`. The quadratic formula states:

For `color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0`, the values of `x` which are the solutions to the equation are given by:

`x = (-color(blue)(b) +- sqrt(color(blue)(b)^2 - (4color(red)(a)color(green)(c))))/(2 * color(red)(a))`

Substituting:

`color(red)(4)` for `color(red)(a)`

`color(blue)(-20)` for `color(blue)(b)`

`color(green)(33)` for `color(green)(c)` gives:

`x = (-(color(blue)(-20)) +- sqrt(color(blue)(-20)^2 - (4 * color(red)(4) * color(green)(33))))/(2 * color(red)(4))`

`x = (color(blue)(20) +- sqrt(400 - 528))/8`

`x = (color(blue)(20) +- sqrt(-128))/8`

`x = (color(blue)(20) +- sqrt(64 * -2))/8`

`x = (color(blue)(20) +- sqrt(64)sqrt(-2))/8`

`x = (color(blue)(20) +- 8sqrt(-2))/8`

Or

`x = color(blue)(20)/8 +- (8sqrt(-2))/8`

`x = 5/2 +- sqrt(-2)`

Answer 2

No Real solutions;
within Complex numbers: `x=2/5-sqrt(2)i" or "2/5+sqrt(2)i`

Explanation:

Given
`color(white)("xxx")(-2x+5)^2=-8`

We note that any Real value squared must be `>= 0`
therefore this equation has No valid Real solutions

If we are dealing with Complex values, then
`color(white)("xxx")(-2x+5)^2=-8`

`color(white)("xxx")4x^2-20x+25=-8`

`color(white)("xxx")4x^2-20x+33=0`

Then applying the quadratic formula that tells us that an equation of the form: `ax^2+bx+c=0`
has solutions:
`color(white)("xxx")x=(-b+-sqrt(b^2-4ac))/(2a)`

We have solutions:
`color(white)("xxx")x=(20+-sqrt(400-4*4*33))/(2 * 4)`

`color(white)("xxx")=(20+-sqrt(-128))/8`

`color(white)("xxx")=(20+-8sqrt(2)i)/8`

`color(white)("xxx")=5/2+-sqrt(2)i`