#1**+1 **

"*Solve y^2 - |4y - 2| + 5 = 0.*" I'm going to assume only real solutions are required.

(1) Suppose 4y-2 is positive. Then we have y^{2}-4y+7=0 but b^{2}-4ac = 16 - 28 <0 so solutions are complex.

(2) Suppose 4y-2 is negative. Then we have y^{2}+4y+3=0. so y = (-4±√(16-12))/2 or y = -1 and y = -3

Check:

If complex solutions are required as well, just solve the quadratic in (1).

Alan Nov 18, 2019

#2**+1 **

I agree with your real solutions, Alan

I also thought that the complex solutions to y^2 - 4y + 7 = 0 were answers

But...Wolframalpha gives these answers for the complex solutions :

https://www.wolframalpha.com/input/?i=y%5E2+-++abs+%284y+-+2+%29++%2B+5+%3D0

Do you see how they got those ???

CPhill Nov 18, 2019

#3**+1 **

Hmm! My simplification when |4y-2| is positive leads to incorrect solutions. i.e. when the solutions are plugged back into the LHS of the original equation the resut isn't zero. The Wolfram Alpha solutions, on the other hand, do give zero, so they are obviously correct. I will need to think further about how to obtain these!

Alan
Nov 18, 2019