Biquadratic Equation Solver
Solve biquadratic equations with the x-squared substitution method, intermediate values, real or complex roots, and a graph of the even polynomial.
A biquadratic equation contains only even powers of the variable. This calculator reduces the input to that form, solves the intermediate equation, and returns real or complex roots.
How a biquadratic equation is solved
A biquadratic equation contains only even powers of the variable. It reduces to an ordinary quadratic after substituting an intermediate variable for x squared.
x is the original unknown; a, b, and c are the biquadratic coefficients.
t is the intermediate variable used before converting back to x.
- Reduce the input to a polynomial with powers 4, 2, and 0
- Substitute t for x squared
- Solve the intermediate quadratic
- Check whether each intermediate value is positive, zero, or negative
- Return real roots and show complex pairs separately
Full equation input
The tool accepts both coefficients and a full equation string. Parentheses, a right-hand side, implicit multiplication, and decimal coefficients are supported as long as the reduced equation remains biquadratic.
x is moved to one side before solving the intermediate quadratic.
x appears only in even powers, so the factored form is valid input.
How many real roots can appear
Nonzero real roots appear in opposite-sign pairs because the polynomial is even. A zero root is shown once in the distinct-root list.
| Intermediate values | Real-root result | Meaning |
|---|---|---|
| Both positive | Four roots | two opposite-sign pairs |
| One positive and one zero | Three distinct roots | one pair plus zero |
| One positive and one negative | Two real roots | negative t gives complex roots |
| Both negative | No real roots | all x roots are complex |
| Intermediate discriminant negative | No real roots | no real t values |
Input limits
If an odd power or a degree higher than four appears after expansion, the problem is no longer biquadratic. The calculator reports that instead of switching to a generic quartic solver.
x appears in a linear term, so the biquadratic structure is broken.
x has degree five, which is outside this solver.
Even symmetry and graph
A biquadratic function is symmetric about the y-axis. The graph helps confirm how many times the curve crosses the x-axis.
f has even symmetry around the y-axis.
Frequently Asked Questions
Sources and References
Calculations are based on the listed reference sources. Links open in a new tab.
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