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

2 Answers
Jul 29, 2017

Answer:

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)#

Jul 29, 2017

Answer:

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#