Question

What are the solutions of the equation x4 – 5x2 – 36 = 0? use factoring to solve.

Answer 1

x^4 - 5x^2 - 36 = 0
(x^2 - 9)(x^2 + 4) = 0
(x-3)(x+3)(x-2)(x+2) = 0

x = -3, -2, 2, 3

Answer 2

x = 3, -3, 2i and -2i where i is an imaginary number.

Explanation:

We can express the equation
`x^4-5x^2-36=0` as
`(x^2)^2-5x^2-36=0`, and we can make
`y = x^2`. So, we have the equation
`y^2-5y-36=0`; to find the solutions of this last equation, we should find two numbers such that these numbers sum up to -5 and multiplied result in -36, these numbers are -9 and 4. Therefore
`y^2-5y-36=0` is equivalent to
`(y-9)(y+4)=0`, but
`y = x^2`, so,
`(x^2-9)(x^2+4)=0`, i.e.,
`(x+3)(x-3)(x^2+4)=0`. This last product is equal to zero when x = 3, x = -3 or
`x^2=-4`,
`x=\pm√(-4)`, i.e.,
`x=\pm2i`. Therefore the solutions of the equation
`x^4-5x^2-36=0` are x = 3, -3, 2i and -2i where i is an imaginary number.