The key to factoring this is to notice (guess and verify) that the square of x^(3/2)x32 is x^3x3
So I can think of this as:
["something"]^2 - 7 ["something"] - 8 = 0[something]2−7[something]−8=0
Until you are more comfortable with the process, do the substitution: the "something" here is x^(3/2)x32 so we'll use a different variable to rename x^(3/2)x32. The traditional variable in this situation is uu.
Let u = x^(3/2)u=x32. That makes u^2 = (x^(3/2))^2 =x^3u2=(x32)2=x3
So we have:
u^2-7u-8=0u2−7u−8=0
(u-8)(u+1)=0(u−8)(u+1)=0
u-8=3u−8=3 or u+1=0u+1=0, so
u=8u=8 or u=-1u=−1
Now go back to the original variable:
x^(3/2)=8x32=8 or x^(3/2) = -1x32=−1
x=8^(2/3)x=823 or x = (-1)^(2/3)x=(−1)23
x=(root(3)8)^2=4x=(3√8)2=4 or x=(root(3)(-1))^2=-1x=(3√−1)2=−1
