Question

Please help!!

Solve the equation and express each solution in a+bi form.
x^4-7x^2-8=0
a. x = -1, 1, 2 sqrt 2, or -2 sqrt 2
b. x = -i, i, 2 sqrt 2, or -2 sqrt 2
c. x = -1, 1, 2 sqrt 2i, or -2 sqrt 2i
d. x = -i, i, 2 sqrt 2i, or -2 sqrt 2i

Answer 1

Because this is a fourth degree that will be tricky to factor as it is, we will do a u substitution. Let
`x^2=u`. We can now rewrite that polynomial in terms of u:
`u^2-7u-8=0`. Filling into the quadratic formula we have
`u=(7+/-√(49-4(1)(-8)))/(2)`. Simplifying down
`u=(7+/-√(81))/(2)` and
`u=(7+9)/(2)` or
`u=(7-9)/(2)`. That means that u = 8 or u = -1. But don't forget that we let
`u=x^2`, so we have to put x-squared back in for u. That gives us
`x^2=8` which simplifies down to
`+/-2√(2)`. That also gives us
`x^2=-1`. When we take the square root of -1, we have to use the fact that -1 = i^2, so we sub that in to get
`x=+/-i`. All in all, your solutions are as follows:
`2√(2),-2√(2),i,-i` which is choice b.