What is the discriminant and why does it matter?

The discriminant (b² - 4ac) determines the nature of the roots. Positive means two distinct real roots; zero means one repeated real root; negative means two complex (imaginary) roots.

What if coefficient a is zero?

If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The calculator will show an error in this case since the quadratic formula requires a ≠ 0.

What are complex roots?

Complex roots occur when the discriminant is negative. They take the form p ± qi where i = √(-1). They appear in conjugate pairs and mean the parabola doesn't cross the x-axis.

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola. For ax² + bx + c, the vertex x-coordinate is -b/(2a) and the y-coordinate is c - b²/(4a).

Can I solve equations where the roots are fractions or irrationals?

Yes. The calculator handles all real and complex solutions, displaying results as decimals. For example, x² - 2 = 0 gives roots ±√2 ≈ ±1.4142136.

Math

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Quadratic Equation Calculator - Solve ax²+bx+c=0 Online

Solve quadratic equations of the form ax² + bx + c = 0. Find real and complex roots using the quadratic formula.

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How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

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In plain English — what does this do?

🔢 This tool does the maths for you! Just type in your numbers, and it gives you the answer right away. No need to count on your fingers or use a pen and paper.

📌 Direct Answer & Summary

A quadratic equation calculator solves equations of the form ax² + bx + c = 0 using the quadratic formula. It finds both real and complex roots, calculates the discriminant, and provides additional properties like the vertex, axis of symmetry, and sum/product of roots.

Enter coefficients for ax² + bx + c = 0

a (x²)

b (x)

c (constant)

What is Quadratic Equation Calculator - Solve ax²+bx+c=0 Online?

How to use it

1️⃣ Enter the coefficients a, b, and c for your equation ax² + bx + c = 0.
2️⃣ The discriminant and roots (x₁ and x₂) appear instantly.
3️⃣ The nature of roots (two real, one repeated, or complex) is shown below the discriminant.
4️⃣ Scroll down to see equation properties including vertex coordinates and axis of symmetry.

Formula

Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Discriminant: Δ = b² - 4ac. If Δ > 0: two real roots. If Δ = 0: one repeated root. If Δ < 0: two complex conjugate roots.

💡 See it in action — a real example

For x² - 5x + 6 = 0 (a=1, b=-5, c=6): Δ = 25 - 24 = 1. Roots: x = (5 ± 1) / 2, giving x₁ = 3 and x₂ = 2. These are the values where the parabola crosses the x-axis.

❓ Common questions

What is the discriminant and why does it matter?: The discriminant (b² - 4ac) determines the nature of the roots. Positive means two distinct real roots; zero means one repeated real root; negative means two complex (imaginary) roots.
What if coefficient a is zero?: If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The calculator will show an error in this case since the quadratic formula requires a ≠ 0.
What are complex roots?: Complex roots occur when the discriminant is negative. They take the form p ± qi where i = √(-1). They appear in conjugate pairs and mean the parabola doesn't cross the x-axis.
What is the vertex of a parabola?: The vertex is the highest or lowest point of the parabola. For ax² + bx + c, the vertex x-coordinate is -b/(2a) and the y-coordinate is c - b²/(4a).
Can I solve equations where the roots are fractions or irrationals?: Yes. The calculator handles all real and complex solutions, displaying results as decimals. For example, x² - 2 = 0 gives roots ±√2 ≈ ±1.4142136.

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