Q: Solve for x:
|x² – 2x| = 3
A: You always get ride of the absolute value signs by breaking an absolute value problem into two equations (a plus version and a minus version):
So,
x² – 2x = 3 or x² – 2x = -3
Now, solve them one at a time:
x² – 2x = 3
x² – 2x – 3 = 0
(x – 3)(x + 1) = 0
x = 3 or -1
And, solve the second equation:
x² – 2x = -3
x² – 2x + 3 = 0
This problem needs to be solved using the quadratic formula (it cannot be factored):
So, x = (-b ± √(b² – 4ac))/(2a)
x =(-(-2) ± √((-2)² – 4*1*3))/(2*1)
x =(2 ± √(4 – 12))/2 = (2 ± √(-8))/2
Now, if you have not studied (or are not studying) imaginary numbers, you would say that there are no solutions at this point because there is a negative number under the square-root, and we cannot take the square-root of a negative number.
If you are studying imaginary numbers, you would continue as follows:
x = (2 ± √(-8))/2 = (2 ± 2√(2)i)/2 = 1 ± √(2)i
Chances are, imaginary numbers should not be included, so your answers are:
x = 3 or -1