Math

Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0 instantly. Enter the coefficients to find real or complex roots with step-by-step explanations.

Enter Coefficients

1x² + 5x + 6 = 0

Try These Examples

x² + 5x + 6 = 0 x² - 3x + 2 = 0 x² - 4 = 0 x² + 2x + 1 = 0 x² + 1 = 0 2x² - 5x + 2 = 0

How to Use the Quadratic Formula

The quadratic formula solves any equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

Understanding the Discriminant

The value under the square root (b² - 4ac) is called the discriminant. It tells you about the roots before you even calculate them:

  • Discriminant > 0: Two distinct real roots
  • Discriminant = 0: One repeated root (both roots are the same)
  • Discriminant < 0: Two complex roots (with imaginary numbers)

Example

Let's solve x² + 5x + 6 = 0:

  1. a = 1, b = 5, c = 6
  2. Discriminant = 5² - 4(1)(6) = 25 - 24 = 1 (positive)
  3. x = (-5 ± √1) / 2 = (-5 ± 1) / 2
  4. x₁ = (-5 + 1) / 2 = -2
  5. x₂ = (-5 - 1) / 2 = -3

So the solutions are x = -2 and x = -3.

Practical Applications

Quadratic equations appear in many real-world situations:

  • Physics: Projectile motion, calculating trajectories
  • Engineering: Bridge design, structural analysis
  • Finance: Profit maximization problems
  • Computer Graphics: Bezier curves, rendering
  • Sports: Optimizing performance metrics

Frequently Asked Questions

What if a = 0?

If a = 0, the equation becomes linear (bx + c = 0) rather than quadratic. Our calculator handles this case too, solving it as a simple linear equation where x = -c/b.

Can I see the steps?

Yes! Every calculation includes a complete step-by-step breakdown showing how the quadratic formula was applied, including the discriminant calculation and each step toward the final answer.

What are complex roots?

When the discriminant is negative, the square root of a negative number involves "i" (the imaginary unit). These are called complex roots and appear in conjugate pairs like a + bi and a - bi.

Is the parabola graph accurate?

The graph shows the parabola y = ax² + bx + c over a reasonable range centered around the vertex. It helps visualize whether the equation opens upward (a > 0) or downward (a < 0) and where the roots appear on the x-axis.