About

Biquadratic equations appear frequently in higher-level algebra, physics, and engineering problems where symmetry plays a role. Unlike standard quartics, these equations lack odd-power terms, allowing them to be solved efficiently using the substitution method. Accuracy in finding these roots is essential for determining equilibrium points in mechanics or intersection points in analytical geometry.

This tool automates the process by transforming the fourth-degree polynomial into a quadratic one. It not only provides the final real and complex roots but also generates a function graph to visualize where the curve intercepts the X-axis, ensuring a comprehensive understanding of the solution set.

Formulas

To solve the equation, we use the substitution method:

ax⁴ + bx² + c = 0

Let t = x². The equation becomes a quadratic:

at² + bt + c = 0

We solve for t using the quadratic formula:

t = −b ± √b² − 4ac2a

Finally, we find x by reversing the substitution:

x = ±√t

Reference Data

Equation Type	Standard Form	Substitution Strategy	Possible Real Roots
Quadratic	ax² + bx + c = 0	None (Direct Formula)	0, 1, or 2
Biquadratic	ax⁴ + bx² + c = 0	Let t = x²	0, 2, or 4
Bicubic (Rare)	ax⁶ + bx³ + c = 0	Let t = x³	0, 1, or 2
General Quartic	ax⁴ + bx³ + cx² + dx + e = 0	Ferrari's Method	0 to 4
Symmetric	ax⁴ + bx³ + cx² + bx + a = 0	Divide by x²	0 to 4
Depressed Quartic	x⁴ + px² + qx + r = 0	Translation	0 to 4
Radical Form	√(ax + b) = cx²	Squaring (leads to quartic)	Varies
Physics Application	Potential Energy Wells	U(x) = x⁴ - x²	Stable/Unstable Equilibrium

Frequently Asked Questions

A biquadratic equation is a fourth-degree polynomial equation that has only even powers of the variable (x^4, x^2) and a constant term. It can be solved like a quadratic equation by substituting x^2 with a temporary variable t.

If the intermediate quadratic equation yields only negative values for t (e.g., t = -4), taking the square root to find x results in imaginary numbers. Since the square of a real number cannot be negative, no real roots exist.

No, this specific method works only for biquadratic forms (ax^4 + bx^2 + c = 0). General quartic equations containing x^3 or x terms require more complex algorithms like Ferrari's method or numerical approximation.

The real roots represent the x-intercepts, or the points where the function's curve crosses the horizontal X-axis. If the curve never crosses the axis, the equation has no real roots.

If b=0, it becomes a pure quartic (ax^4 + c = 0). If c=0, you can factor out x^2, leaving a standard quadratic. The solver handles these cases automatically, provided 'a' is not zero.