Quadratic Equation Solver – Solve ax² + bx + c = 0 Online

Q: What is a quadratic equation?

A quadratic equation is ax² + bx + c = 0 where a ≠ 0. The highest power of the variable is 2.

Q: What is the discriminant?

The discriminant (Δ = b² − 4ac) determines the nature of the roots. Δ > 0: two real roots. Δ = 0: one repeated root. Δ < 0: two complex roots.

Q: What are complex roots?

When discriminant is negative, roots involve imaginary numbers expressed as a + bi where i² = -1.

Q: Is this Quadratic Solver free?

Yes, completely free with no signup required.

Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0 instantly. Get roots, discriminant, vertex, and step-by-step solution.

Quick Examples

Enter Coefficients

Standard Form

ax² + bx + c = 0

a (x²)b (x)c (constant)

Quadratic Formula

x = (−b ± √(b²−4ac)) ÷ 2a

Solution

Enter coefficients a, b, c and click Solve.

Quadratic Equation Solver — Complete Solution

A quadratic equation is any equation in the form ax² + bx + c = 0 where a ≠ 0. Our solver uses the quadratic formula to find all roots instantly — whether they are two real roots, one repeated root, or two complex roots. It also calculates the discriminant, vertex, axis of symmetry, and provides a full step-by-step solution.

Quadratic equations appear everywhere — from calculating projectile motion in physics, to finding profit-maximizing quantities in economics, to solving geometry problems. Our calculator handles all three cases automatically based on the value of the discriminant.

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All Root Types

Handles two real roots, one repeated root, and complex roots — all cases covered automatically.

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Full Properties

Get discriminant, vertex, axis of symmetry, sum and product of roots, and parabola direction.

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Step-by-Step

Shows the complete solution process — perfect for students learning how to solve quadratic equations.

The Quadratic Formula

The most important formula in algebra

Quadratic Formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Δ > 0

Two real roots

Δ = 0

One repeated root

Δ < 0

Two complex roots

Understanding the Discriminant

The discriminant (Δ = b² − 4ac) tells you exactly what kind of roots the equation has before you even solve it. It is the expression inside the square root in the quadratic formula.

Δ > 0 — Two Distinct Real Roots

The parabola crosses the x-axis at two points. Example: x² − 5x + 6 = 0 has roots x = 2 and x = 3.

Δ = 0 — One Repeated Root

The parabola just touches the x-axis at one point (vertex). Example: x² + 4x + 4 = 0 has root x = −2 (repeated).

Δ < 0 — Two Complex Roots

The parabola does not cross the x-axis. Roots involve imaginary numbers. Example: x² + x + 1 = 0 has complex roots.

Key Properties Reference

Property	Formula
Discriminant	b² − 4ac
Root 1	(−b + √Δ) ÷ 2a
Root 2	(−b − √Δ) ÷ 2a
Vertex x	−b ÷ 2a
Vertex y	c − b² ÷ 4a
Axis of Symmetry	x = −b ÷ 2a
Sum of Roots	−b ÷ a
Product of Roots	c ÷ a

Worked Examples

x² − 5x + 6 = 0

a=1, b=-5, c=6

Δ = b²−4ac = 25 − 24 = 1

x₁ = 3, x₂ = 2

Two real roots

x² + 4x + 4 = 0

a=1, b=4, c=4

Δ = b²−4ac = 16 − 16 = 0

x = −2 (repeated)

One repeated root

x² + x + 1 = 0

a=1, b=1, c=1

Δ = b²−4ac = 1 − 4 = −3

x = −0.5 ± 0.866i

Complex roots

2x² − 3x − 2 = 0

a=2, b=-3, c=-2

Δ = b²−4ac = 9 + 16 = 25

x₁ = 2, x₂ = −0.5

Two real roots

Real-World Applications

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Projectile Motion

Calculate when a thrown object hits the ground. The height equation h = −16t² + vt + h₀ is quadratic in time t.

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Geometry

Find the dimensions of a rectangle given its area. If length × width = area and length − width = difference, the result is quadratic.

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Economics

Maximize profit or minimize cost using quadratic revenue and cost functions. The profit-maximizing quantity is the vertex of the parabola.

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Physics & Optics

Lens equations, satellite dish parabolas, and beam deflection problems all involve quadratic equations.

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Engineering

Structural load calculations, cable tension in suspension bridges, and stress analysis use quadratic models.

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Statistics

Quadratic regression fits a parabola to data points, useful when the relationship between variables is non-linear.

Quadratic Solver FAQ

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0 where a ≠ 0. The highest power of the variable is 2, which is why it is called "quadratic" (from the Latin for square).

What is the discriminant?

The discriminant (Δ = b² − 4ac) determines the nature of the roots. If Δ > 0, two real roots. If Δ = 0, one repeated root. If Δ < 0, two complex roots with no real solution.

What are complex roots?

When the discriminant is negative, the square root of a negative number appears in the formula. These are expressed using the imaginary unit i (where i² = −1), giving complex numbers like 2 + 3i.

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola. Its x-coordinate is −b÷2a and y-coordinate is c − b²÷4a. If a > 0 it is the minimum point; if a < 0 it is the maximum point.

What if a = 0?

If a = 0, the equation becomes linear (bx + c = 0), not quadratic. Our solver requires a ≠ 0 and will show an error if you enter a = 0.

Is this Quadratic Solver free?

Yes, completely free with no signup or registration required. Use it as many times as you need.