Roots of Biquadratic Equation Calculator

A biquadratic equation is a type of polynomial equation that can be solved using substitution. This calculator helps you find all real roots of a biquadratic equation in the form ax⁴ + bx² + c = 0.

What is a Biquadratic Equation?

A biquadratic equation is a fourth-degree polynomial equation that contains only even powers of x. The general form is:

ax⁴ + bx² + c = 0

Where a, b, and c are constants, and a ≠ 0. The equation is called "biquadratic" because it resembles a quadratic equation but with x² substituted for x.

Unlike general quartic equations, biquadratic equations have a special property that allows them to be solved using substitution, making them simpler to solve than other fourth-degree equations.

How to Solve a Biquadratic Equation

To solve the equation ax⁴ + bx² + c = 0, follow these steps:

Divide the entire equation by a to simplify it to x⁴ + (b/a)x² + (c/a) = 0.
Let y = x². This substitution transforms the equation into a quadratic equation in terms of y: y² + (b/a)y + (c/a) = 0.
Solve the quadratic equation for y using the quadratic formula: y = [-b/a ± √((b/a)² - 4(c/a))]/2.
Check if the discriminant (D = (b/a)² - 4(c/a)) is positive, zero, or negative to determine the nature of the roots.
If D ≥ 0, find the real roots by taking the square roots of the positive y values: x = ±√y.
If D < 0, there are no real roots (the roots are complex).

Note: The substitution method works because the equation is biquadratic, meaning it only contains even powers of x. This allows us to treat it as a quadratic equation in terms of y = x².

Example Calculation

Let's solve the equation 2x⁴ - 3x² - 5 = 0 using our calculator.

Step-by-Step Solution

Divide by 2: x⁴ - (3/2)x² - 5/2 = 0
Let y = x²: y² - (3/2)y - 5/2 = 0
Solve quadratic equation: y = [3/2 ± √((9/4) + 10)]/2 = [3/2 ± √(49/4)]/2 = [3/2 ± 7/2]/2
Solutions for y: y₁ = (3/2 + 7/2)/2 = 5, y₂ = (3/2 - 7/2)/2 = -2
Find x: x = ±√5, x = ±√(-2) (complex roots)
Real roots: x = ±√5 ≈ ±2.236

The calculator will show these results along with a graphical representation of the roots.

Frequently Asked Questions

What is the difference between a biquadratic and a quadratic equation?: A quadratic equation has the form ax² + bx + c = 0, while a biquadratic equation has the form ax⁴ + bx² + c = 0. The biquadratic equation can be solved by substitution with y = x².
Can a biquadratic equation have complex roots?: Yes, if the discriminant of the transformed quadratic equation is negative, the biquadratic equation will have complex roots.
How many real roots can a biquadratic equation have?: A biquadratic equation can have 0, 2, or 4 real roots. The number depends on the discriminant of the transformed quadratic equation.
Is the substitution method the only way to solve a biquadratic equation?: Yes, because of the special form of the biquadratic equation, substitution with y = x² is the most straightforward method.