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Quadratic Equatrion Roots Calculator

Reviewed by Calculator Editorial Team

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form ax² + bx + c = 0. The roots of the equation are the values of x that satisfy the equation. This calculator uses the quadratic formula to find the roots of any quadratic equation.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2. It has the general form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0.

The graph of a quadratic equation is a parabola. The roots of the equation are the points where the parabola intersects the x-axis. A quadratic equation can have:

  • Two distinct real roots
  • One real root (a repeated root)
  • No real roots (the roots are complex numbers)

The nature of the roots is determined by the discriminant, which is calculated as b² - 4ac.

How to Solve Quadratic Equations

The Quadratic Formula

The standard method for solving quadratic equations is the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

This formula provides the two possible solutions for x. The discriminant (b² - 4ac) determines the nature of the roots:

  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is exactly one real root.
  • If the discriminant is negative, there are two complex conjugate roots.

Example Calculation

Let's solve the equation x² - 5x + 6 = 0:

  1. Identify the coefficients: a = 1, b = -5, c = 6
  2. Calculate the discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1
  3. Apply the quadratic formula:
    • x = [5 ± √1] / 2
    • x = (5 + 1)/2 = 3
    • x = (5 - 1)/2 = 2

The roots are x = 2 and x = 3.

Special Cases

When a = 0, the equation is linear, not quadratic. When b = 0, the equation is a pure quadratic. When c = 0, the equation has roots at x = 0 and x = -b/a.

Real-World Applications

Quadratic equations are used in various real-world scenarios, including:

  • Projectile motion in physics
  • Calculating areas and volumes in geometry
  • Optimization problems in business and economics
  • Modeling growth and decay in biology and chemistry

For example, in physics, the height of a projectile can be modeled using a quadratic equation where the roots represent the times when the projectile hits the ground.

Frequently Asked Questions

What is the discriminant in a quadratic equation?
The discriminant is the part of the quadratic formula under the square root (b² - 4ac). It determines the nature of the roots: positive for two real roots, zero for one real root, and negative for complex roots.
Can a quadratic equation have complex roots?
Yes, if the discriminant is negative, the roots will be complex numbers. These are solutions in the form of a + bi, where i is the imaginary unit.
What happens when a = 0 in a quadratic equation?
When a = 0, the equation becomes linear (bx + c = 0), not quadratic. It will have exactly one solution unless b is also zero.
How do I know if a quadratic equation has real roots?
A quadratic equation has real roots if the discriminant (b² - 4ac) is positive. If the discriminant is zero, there is exactly one real root. If negative, the roots are complex.