Solve the equation: (3-8x^2)^(1/4) = 2x?

1 Answer
Jan 29, 2015

Raise both sides to the 4th power:
((3-8x^2)^(1/4))^4 = (2x)^4

Simplify:
3-8x^2 = 2^4*x^4
3-8x^2 = 16x^4
0 = 16x^4+8x^2-3
0 = (4x^2 - 1)(4x^2 + 3)

So: 4x^2-1 = 0 or 4x^2+3=0

4x^2-1 = 0 -> 4x^2=1 -> x^2 = 1/4 -> x = +- 1/2
4x^2+3=0 -> 4x^2=-3 -> not a real solution

Now we have to check for extraneous solutions:
x=1/2:
Left side: (3-8*(1/4))^(1/4) = (3-2)^(1/4) = 1^(1/4) = 1
Right side: 2*1/2 = 1
Left and right side are equal, so this solution works

x=-1/2:
Left side: (3-8*(1/4))^(1/4) = (3-2)^(1/4) = 1^(1/4) = 1
Right side: 2*-1/2 = -1
Left and right side are not equal, so this solution is extraneous.

So our answer: x=1/2