📐 Math Tool

Quadratic FormulaCalculator

x= (−b ± √(b²−4ac)) / 2a

Solve any quadratic equation step-by-step. Get real and complex roots instantly — no sign-up, completely free.

✦ Real & Complex Roots✦ Step-by-Step Working✦ 100% Free✦ No Sign-up
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Quadratic Formula Calculator

Enter coefficients a, b, and c for ax² + bx + c = 0

Your Equation
1x² +0x +0= 0
Coefficient a
a ≠ 0
Coefficient b
can be 0
Coefficient c
constant term
Try an example:

Solution

Step-by-Step

Enter coefficients and click
Solve Equationto see
roots with step-by-step working.

📖 The Quadratic Formula

Aquadratic equationhas the formax² + bx + c = 0, where a ≠ 0. The quadratic formula gives the exact solution for any values of a, b, and c:

x = (−b ± √(b²−4ac)) / 2a

The±symbol means there are (usually) two solutions — one using addition and one using subtraction.

🔍 Understanding the Discriminant

Thediscriminant(Δ = b²−4ac) tells you how many real roots exist before you solve:

  • Δ > 0:Two distinct real roots
  • Δ = 0:One repeated real root (perfect square)
  • Δ < 0:Two complex (imaginary) roots — no real solutions

Complex roots always come in conjugate pairs:p ± qi

🔄 Alternative Methods

Besides the quadratic formula, you can also solve quadratics by:

  • Factoring:Express as (x − r₁)(x − r₂) = 0 when roots are rational
  • Completing the Square:Rewrite as (x + p)² = q
  • Graphing:Find where the parabola y = ax² + bx + c crosses the x-axis

The quadratic formula always works — even when factoring fails.

📊 Real-World Applications

  • Physics:Projectile motion (finding when/where objects land)
  • Engineering:Structural load calculations
  • Economics:Finding break-even points (profit = revenue − costs)
  • Architecture:Parabolic arch design
  • Finance:Compound interest growth models
  • Computer graphics:Bezier curve intersections

Frequently Asked Questions

What if a = 0?
If a = 0, the equation is no longer quadratic — it becomes linear:bx + c = 0. The quadratic formula requires a ≠ 0 (you'd be dividing by 2a = 0, which is undefined). This calculator will flag this error.
What does it mean to have complex roots?
When the discriminant is negative (Δ < 0), the square root of a negative number involvesi(imaginary unit, where i² = −1). The roots are complex numbers in the formp ± qi. In practical terms, this means the parabola doesn't intersect the x-axis — there are no real solutions.
What is a "repeated root"?
When the discriminant equals 0, both roots are identical:x = −b / 2a. This is called a repeated (or double) root. The parabola touches the x-axis at exactly one point — its vertex.
Can I use this for decimals and fractions?
Yes! Enter any real number for a, b, and c — including decimals like0.5,1.25, or negatives like-3.7. For fractions, convert to decimals first (e.g., ½ = 0.5). The calculator handles all real-number inputs.
How do I verify my roots are correct?
Substitute each root back into the original equation. If a·x² + b·x + c = 0 (or very close to 0 for decimal answers), the root is correct. You can also verify using Vieta's formulas: the sum of roots equals−b/aand the product equalsc/a.