Biquadratic Equations
The calculator solves the biquadratic equation ax⁴ + bx² + c = 0 with the substitution t = x², step by step.
Enter the coefficients (an empty field means zero):
x⁴ +x² += 0
The biquadratic equation solver solves ax⁴ + bx² + c = 0 with the substitution t = x² and shows every step: the quadratic equation in t, choosing the non-negative roots and substituting back.
What is a biquadratic equation
A biquadratic equation is a fourth-degree equation that contains only even powers of x:
ax4 + bx2 + c = 0, a ≠ 0
How to solve it
- Substitute t = x2, with t ≥ 0. You get the quadratic equation at2 + bt + c = 0.
- Solve it using the discriminant.
- Discard negative values of t — a square is never negative.
- For each t > 0 find x = ±√t; for t = 0, x = 0.
Example. x4 − 5x2 + 4 = 0
t = x2: t2 − 5t + 4 = 0, t1 = 1, t2 = 4
x2 = 1 ⇒ x = ±1; x2 = 4 ⇒ x = ±2
Answer: x = −2; −1; 1; 2
How many roots there can be
| Roots of the equation in t | Roots in x |
|---|---|
| two positive | four |
| one positive, one negative | two |
| one of them is zero | x = 0 plus two more or none |
| both negative, or D < 0 | no real roots |