Quadratic Formula Calculator

Q: Can a equal zero in a quadratic equation?

No. If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0.

Q: How do I find the vertex of a parabola from a, b, and c?

The vertex x-coordinate is -b/2a. Substitute that value back into the original equation to get the corresponding y-coordinate, c - b²/4a.

Solve any quadratic equation instantly with our free quadratic formula calculator. Shows the discriminant, real or complex roots, vertex, and axis of symmetry with full step-by-step working.

What is Quadratic Formula Calculator?

A quadratic equation has the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It's called "quadratic" because the highest power of the variable x is 2. Quadratic equations describe parabolas — U-shaped curves — and appear throughout algebra, physics, engineering, and everyday problem-solving, from calculating projectile trajectories to optimizing areas and profits.

This calculator uses the quadratic formula, one of the most important results in algebra, to find the exact roots (solutions) of any quadratic equation, whether those roots are real numbers or complex numbers.

Formula

x = (-b ± √(b² - 4ac)) / 2a

Discriminant: Δ = b² - 4ac

If Δ > 0: two distinct real roots

If Δ = 0: one repeated real root

If Δ < 0: two complex conjugate roots

Vertex: x = -b/2a, y = c - b²/4a

How to Calculate

Enter the coefficients a, b, and c from your equation ax² + bx + c = 0.
The calculator computes the discriminant (b² - 4ac) to determine the type of roots.
The quadratic formula is applied to find the exact root values.
Review the full step-by-step solution, plus the vertex and axis of symmetry.

Example

For x² - 3x + 2 = 0 (a=1, b=-3, c=2): the discriminant is (-3)² - 4(1)(2) = 9 - 8 = 1. Since Δ > 0, there are two real roots: x = (3 ± 1) / 2, giving x₁ = 2 and x₂ = 1.

Key Benefits

Instant, exact roots for any quadratic equation
Automatically handles real and complex roots
Full step-by-step working, not just the final answer
Also shows the vertex and axis of symmetry

Common Mistakes to Avoid

Forgetting the ± sign, which produces only one of the two roots
Mixing up the signs of b and c when substituting into the formula
Assuming a negative discriminant means "no solution" instead of "complex solutions"
Dividing by 2a incorrectly when a is negative

Pro Tips

If a, b, and c are all divisible by a common factor, simplify the equation first
Check your answer by substituting each root back into the original equation
A quick way to estimate real-root sign: if a and c have opposite signs, the discriminant is guaranteed positive
The sum of the roots is always -b/a, and the product is always c/a — useful for a fast sanity check

Key Terms Explained

Discriminant: The expression b² - 4ac, which determines the nature of the roots
Root: A value of x that satisfies the equation, also called a zero or solution
Vertex: The turning point of the parabola, either its minimum or maximum
Axis of Symmetry: The vertical line x = -b/2a that the parabola is mirrored across

Common Use Cases

Solving algebra homework and checking your work
Finding the trajectory peak and landing point of a projectile in physics
Optimizing area, profit, or revenue functions in business and engineering
Preparing for standardized tests like the SAT, ACT, or GCSE

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves any equation of the form ax² + bx + c = 0 by giving both roots directly from the coefficients a, b, and c.

What does a negative discriminant mean?

A negative discriminant (b² - 4ac < 0) means the equation has no real roots — the parabola never touches the x-axis. Instead, it has two complex conjugate roots of the form p ± qi.

Can a equal zero in a quadratic equation?

No. If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0.

How do I find the vertex of a parabola from a, b, and c?

The vertex x-coordinate is -b/2a. Substitute that value back into the original equation to get the corresponding y-coordinate, c - b²/4a.

Related Tools

Percentage Calculator

Scientific Calculator

What is Quadratic Formula Calculator?

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves any equation of the form ax² + bx + c = 0 by giving both roots directly from the coefficients a, b, and c.

What does a negative discriminant mean?

A negative discriminant (b² - 4ac < 0) means the equation has no real roots — the parabola never touches the x-axis. Instead, it has two complex conjugate roots of the form p ± qi.

Can a equal zero in a quadratic equation?

No. If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0.

How do I find the vertex of a parabola from a, b, and c?

The vertex x-coordinate is -b/2a. Substitute that value back into the original equation to get the corresponding y-coordinate, c - b²/4a.