Biquadratic Equation Solver
Solve biquadratic equations of the form ax⁴ + bx² + c = 0 using the substitution t = x². Get the discriminant, intermediate roots, and all real solutions step by step.
ax⁴ + bx² + c = 0
How to Use the Biquadratic Equation Solver
- Enter the numbers or values in the input fields.
- The result is calculated and displayed automatically.
- Review the step-by-step solution or detailed breakdown.
- Copy the result or adjust inputs for a new calculation.
Quick Reference
| From | To |
|---|---|
| x² + 5x + 6 = 0 | x = −2, −3 |
| x² − 4 = 0 | x = 2, −2 |
| 2x² − 3x − 2 = 0 | x = 2, −0.5 |
| x² − 1 = 0 | x = 1, −1 |
| x² + 2x + 1 = 0 | x = −1 |
| x² − 5x + 6 = 0 | x = 2, 3 |
Use Cases
- •Checking homework or exam answers quickly and accurately.
- •Verifying manual calculations in professional or academic work.
- •Learning mathematical concepts with instant visual feedback.
- •Performing quick computations during meetings or presentations.
Formula
A biquadratic equation ax⁴ + bx² + c = 0 is solved by substituting t = x², yielding the quadratic at² + bt + c = 0. Solve for t using the quadratic formula, then find x = ±√t for each positive value of t. Up to 4 real roots are possible.
Frequently Asked Questions
What is a biquadratic equation?
A biquadratic equation has the form ax⁴ + bx² + c = 0. It contains only even powers of x, which allows it to be reduced to a quadratic equation by substituting t = x².
How many roots can a biquadratic equation have?
A biquadratic equation can have 0, 1, 2, 3, or 4 real roots. The number depends on the discriminant and the signs of the intermediate t values.
What if both t values are negative?
If both solutions for t are negative, then there are no real roots for x, since x² cannot be negative in real numbers.
Can a be zero?
No. If a = 0, the equation is no longer biquadratic — it becomes a standard quadratic bx² + c = 0 instead.