Question

Find all the roots of the equation x ⁴ −6x ² −8=0.

A) x=−2,x=2
B) x=−2,x=−1,x=1,x=2
C) x=−4,x=4
D) x=−3,x=1

Answer 1

The equation x⁴ - 6x² - 8 = 0 can be solved by substituting u = x² and factoring the resultant quadratic equation. The solutions for u are 8 and -1, corresponding to real roots x = -2√2 and x = 2√2 after reverting the substitution and taking square roots.

Step-by-step explanation:

We are given the equation x⁴ − 6x² − 8 = 0 and we need to find all the roots of this equation. To start solving this, we notice that the equation resembles a quadratic in form if we introduce a substitution. Let u = x², then the equation becomes u² - 6u - 8 = 0. This is a quadratic equation that can be solved by the quadratic formula or factoring.

First, we can factor the equation: (u - 8)(u + 1) = 0. This means u = 8 or u = -1. Since u = x², we take the square root of each solution: For u = 8, x = ±2√8, and for u = -1, the solution is not a real number because we cannot take the square root of a negative number in the real number system.

Hence, x = ±2√8, which simplifies to x = -2√2 or x = 2√2. We express √8 as 2√2 to simplify the roots. Therefore, the roots of the equation are x = -2√2, x = 2√2.