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Solve for Roots in Quadratic Equation Calculator

Reviewed by Calculator Editorial Team

Quadratic equations are fundamental in algebra and appear in many real-world problems. This calculator helps you find the roots of any quadratic equation in the standard form ax² + bx + c = 0.

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x with the general form:

ax² + bx + c = 0

Where:

  • a, b, and c are constants
  • a ≠ 0 (if a = 0, the equation is linear, not quadratic)
  • x is the variable we solve for

Quadratic equations can represent many real-world situations, such as projectile motion, area problems, and optimization tasks.

The Quadratic Formula

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

x = [-b ± √(b² - 4ac)] / (2a)

This formula will always give two solutions for x, called roots or zeros of the equation. The solutions may be:

  • Two distinct real numbers
  • One real number (a repeated root)
  • Two complex numbers

The formula works for any quadratic equation as long as a, b, and c are real numbers and a ≠ 0.

The Discriminant

The discriminant is the part of the quadratic formula under the square root (b² - 4ac). It determines the nature of the roots:

Discriminant (D) = b² - 4ac

The discriminant tells us:

  • If D > 0: Two distinct real roots
  • If D = 0: One real root (repeated)
  • If D < 0: Two complex roots

This calculator will show you the discriminant along with the roots.

How to Use This Calculator

  1. Enter the coefficients a, b, and c of your quadratic equation
  2. Click "Calculate Roots"
  3. View the results including the roots and discriminant
  4. Interpret the results based on the discriminant value

Note: For complex roots, the calculator will show them in the form a + bi where i is the imaginary unit (√-1).

Example Calculation

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

  1. Identify coefficients: a = 1, b = -5, c = 6
  2. Calculate discriminant: D = (-5)² - 4(1)(6) = 25 - 24 = 1
  3. Apply quadratic formula:
    x = [5 ± √1] / 2
  4. Find roots:
    • x₁ = (5 + 1)/2 = 3
    • x₂ = (5 - 1)/2 = 2

The roots are x = 2 and x = 3.

FAQ

What if a = 0 in the quadratic equation?
The equation becomes linear (bx + c = 0) and has only one solution: x = -c/b. This calculator only works for quadratic equations where a ≠ 0.
Can quadratic equations have complex roots?
Yes, when the discriminant is negative (D < 0), the roots are complex numbers. The calculator will display them in the form a + bi.
What does it mean if the discriminant is zero?
A discriminant of zero means there's exactly one real root (a repeated root). The parabola touches the x-axis at one point.
How accurate is this calculator?
The calculator uses standard floating-point arithmetic which is accurate to about 15 decimal places. For most practical purposes, this is sufficient.
Can I use this calculator for higher-degree polynomials?
No, this calculator is specifically designed for quadratic equations (degree 2). For higher-degree polynomials, you would need a different type of calculator.