Question

Suppose that `g(x)=5x^4-15x^2-32`. How do you solve the equation for `x` if `g(x)=-32`? What about `g(x)=58`?

Answer 1

Case 1: `g(x)=-32 rarr color(green)(x in {0,+-sqrt(93)})`

Case 2: `g(x)=58 rarr color(green)(x in {+-sqrt(6),+-sqrt(3)i})`

Explanation:

Given: `color(blue)(g(x)=5x^4-15x^2-32`

Part 1: `color(red)("If "g(x)=-32)`

`color(red)(-32)=color(blue)(5x^4-15x^2-32)`

`rarr color(blue)(5x^4-15x^2)=0`

`rarr 5xxx^2xx(x^2-3)=0`

#rarr{(x^2=0,color(white)("X")orcolor(white)("X"),x^2-3=0), (rarrx=0,,rarrx=+-sqrt(3)) :}#

`x in {-sqrt(3),0,+sqrt(3)}`

Part 2: `color(red)("If "g(x)=58)`

`color(red)(58)=color(blue)(5x^4-15x^2-32)`

`rarr color(blue)("5x^4-15x^2)-90=0`

`rarr 5xx(x^2-6)xx(x^2+3)=0`

#rarr{((x^2-6)=0,color(white)("X")orcolor(white)("X"),x^2+3=0), (rarrx=+-sqrt(6),,rarrx=+-sqrt(3)i) :}#

`x in{-sqrt(6),+sqrt(6),-sqrt(3)i,+sqrt(3)i}`