How to solve equations of the form ax^4 + bx^2 +c = 0?

Given: ax^4 + bx^2 +c = 0

Let u = x^2.

Substitute this variable u into the given equation.

au^2 + bu + c = 0 --> quadratic equation in variable u.

Then solve it the usual way like you do with quadratic equations.

BUT remember that u = x^2 so once you've found u, you still need to take the square root to find x.

The trick here is to create another variable, say 'y', where y = x^2

ax^4 + bx^2 + c = 0

a(x^2)^2 + b(x^2) + c = 0

a(y)^2 + b(y) + c = 0

You are now left with a quadradic equation.

Let t = x² then you have at²+bt+c = 0 and the quadratic formula is readily applicable.

This is a special form of the quadratic. A quadratic equation can be more correctly defined as:

a * f(x)^2 + b * f(x) + c = 0

f(x) = (-b +/- sqrt(b^2 - 4ac)) / (2a)

In this case, f(x) = x^2

x^2 = (-b +/- sqrt(b^2 - 4ac)) / (2a)

x = +/- sqrt((-b +/- sqrt(b^2 - 4ac)) / (2a))

This is EXACTLY the type of equation that is solved by the quadratic formula!

x = [-b ± √(b²-4ac)] / (2a)