Roots of Bi Quadratic Equation Calculator

A bi quadratic equation is a fourth-degree polynomial equation of the form ax⁴ + bx² + c = 0. This calculator helps you find all real roots of such equations by transforming them into quadratic equations through substitution.

What is a Bi Quadratic Equation?

A bi quadratic equation is a special type of polynomial equation that can be written in the form:

ax⁴ + bx² + c = 0

Where:

a, b, and c are coefficients
x is the variable
a ≠ 0 (since it's a fourth-degree equation)

These equations are called "bi quadratic" because they can be factored into two quadratic equations. The solutions to the bi quadratic equation are the same as the solutions to these two quadratic equations.

How to Solve a Bi Quadratic Equation

To solve a bi quadratic equation of the form ax⁴ + bx² + c = 0, follow these steps:

Divide the entire equation by the coefficient of x⁴ (a) to make it monic:
x⁴ + (b/a)x² + (c/a) = 0
Let y = x². This substitution transforms the equation into a quadratic 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
For each real solution y, find x by taking the square root of y:
x = ±√y

This method gives all real roots of the original bi quadratic equation.

Note: If the discriminant (b/a)² - 4(c/a) is negative, the equation has no real roots. If it's zero, there's one real double root. If positive, there are two distinct real roots.

Example Calculation

Let's solve the equation x⁴ - 5x² + 4 = 0:

Divide by the coefficient of x⁴ (which is 1 in this case):
x⁴ - 5x² + 4 = 0
Let y = x²:
y² - 5y + 4 = 0
Solve the quadratic equation:
y = [5 ± √(25 - 16)] / 2 = [5 ± 3]/2

This gives two solutions for y:
- y₁ = (5 + 3)/2 = 4
- y₂ = (5 - 3)/2 = 1
Find x for each y:
- For y₁ = 4: x = ±√4 = ±2
- For y₂ = 1: x = ±√1 = ±1

The roots of the equation x⁴ - 5x² + 4 = 0 are x = -2, -1, 1, and 2.

Frequently Asked Questions

What is the difference between a quadratic and a bi quadratic equation?

A quadratic equation is of the form ax² + bx + c = 0, while a bi quadratic equation is of the form ax⁴ + bx² + c = 0. Bi quadratic equations can be solved by transforming them into quadratic equations through substitution.

How many roots can a bi quadratic equation have?

A bi quadratic equation can have up to four real roots, depending on the discriminant of the transformed quadratic equation. It can have no real roots, two real roots, or four real roots.

Can all bi quadratic equations be solved using the method described?

Yes, the method described works for all bi quadratic equations of the form ax⁴ + bx² + c = 0. It transforms the equation into a quadratic form that can be solved using the quadratic formula.