solve the equation by making an appropriate substitution

Question

x⁴ -11x²+16=0

let u=4, then the quadratic equation in u is ?

Answer 1

A good substitution is

u = x² ⇒

u² - 11u + 16 = 0

Solve for u using the quadratic formula. Note the discriminant is positive, so you will get 2 different real solutions for u. Then evaluate

x = √u.

This gives 2 solutions for each u, so you will have 4 solutions for x, as you would expect from the degree 4 polynomial in x.