Quadratic Formula Calculator

Last updated: Jul 22, 2026 | Reviewed by Calkulator Team | 2 min read

$Math illustration$

Math

Solve quadratic equations and find roots instantly

For ax² + bx + c = 0, the discriminant determines the nature of roots. Positive discriminant means two real roots, zero means one repeated root, negative means complex roots. Enter coefficients and see the complete solution.

The vertex of a parabola is at x = -b/2a — this is the minimum or maximum point of the quadratic function.

Coefficients

ax² + bx + c = 0

a (coefficient of x²)

Cannot be 0 (would make it linear)

b (coefficient of x)

c (constant term)

Formula

x = (−b ± √(b²−4ac)) / 2a
Discriminant (D) = b² − 4ac
D > 0 → 2 real roots
D = 0 → 1 repeated root
D < 0 → 2 complex roots

Roots

x₁ = —

x₂ = —

Discriminant (D)—

Nature of Roots—

Vertex (h, k)—

Axis of Symmetry—

Live Result Illustration

Visual summary — updates instantly as you enter values above

LIVE

Real-Life Guide to Using the Quadratic Formula

Roots of ax² + bx + c = 0. Use the examples and checks below to turn the number into a practical decision.

When this calculator is useful

Use this to find the roots of an equation in the form ax² + bx + c = 0 when factoring is not obvious, or to check the discriminant and determine whether the roots are real or complex.

For most people, the best way to use the Quadratic Formula is to try the real case first, then change one input at a time. That makes the trade-off visible. For example, with a loan calculator you can change tenure while keeping the same rate; with an investment calculator you can change return assumption while keeping the same monthly contribution; with a health, education or measurement calculator you can check how much one input changes the final category.

The result should answer a practical question: Can I afford this? How much should I save? Is this score enough? Is this measurement within range? What is the safer or cheaper option? If the output does not answer the decision clearly, adjust the inputs until the scenario matches your real situation.

Real-Life Example

A projectile height problem: A ball's height over time follows h = -5t² + 20t + 1, and the time when it returns to height 0 is needed.

1For -5t² + 20t + 1 = 0: a = -5, b = 20, c = 1. Discriminant = 20² - 4(-5)(1) = 400 + 20 = 420, and √420 ≈ 20.49. So t = (-20 ± 20.49) ÷ (2 × -5), giving t ≈ -0.049 or t ≈ 4.05.

2Now change one input, such as rate, time, quantity, unit or score, and compare the new result with the first one.

When a quadratic models a real situation, discard any root that does not make physical sense, such as negative time, keeping only t ≈ 4.05 seconds.

Practical Advice

Use the Quadratic Formula as a planning tool, not just a number generator. Write down the inputs you used, because the final answer is meaningful only when you remember the assumptions behind it.

If the decision affects money, health, tax, safety, academics or legal compliance, keep a second check ready. That second check may be a bank quote, payslip, official rule, prescription, site measurement, mark sheet or invoice.

Common Mistakes

Forgetting the ± in the formula and reporting only one root instead of both x = (-b + √D) ÷ 2a and x = (-b - √D) ÷ 2a.
Making a sign error in the discriminant b² - 4ac when a or c is negative — for x² - 4x - 5 = 0, the discriminant is 16 - 4(1)(-5) = 16 + 20 = 36, not 16 - 20.
Dividing only part of the numerator by 2a, computing (-b + √D) ÷ 2a as -b + (√D ÷ 2a) instead of dividing the entire sum by 2a.
Assuming a negative discriminant means "no solution" rather than "no real solution" — the equation still has two complex conjugate roots.
Not reading off a, b, and c correctly from a non-simplified equation, such as failing to recognize a = 2, b = 8, c = 6 directly from 2x² + 8x + 6 = 0.

How to Interpret Results

Check the discriminant value alongside the roots — a positive discriminant means two distinct real roots, zero means one repeated real root, and a negative discriminant means two complex conjugate roots with no real x-intercepts on a graph.

A good interpretation looks at both the main result and the supporting values. If a page shows totals, ratios, categories, schedules or warnings, read those together instead of focusing only on the biggest number.

Quadratic Formula FAQs

Useful answers for interpreting the output, avoiding mistakes and using the result responsibly.

What does this calculator solve for?

It finds the value(s) of x that satisfy any equation in the standard quadratic form ax² + bx + c = 0, returning either two real roots, one repeated root, or two complex roots.

What is the quadratic formula itself?

x = (-b ± √(b² - 4ac)) ÷ 2a, where a, b, and c are the coefficients of the x², x, and constant terms respectively.

What does the discriminant tell you?

The discriminant, b² - 4ac, determines the nature of the roots: positive gives two distinct real roots, exactly zero gives one repeated real root, and negative gives two complex conjugate roots.

Why not just factor the quadratic instead?

Factoring works quickly for simple integer roots, but many quadratics have irrational or complex roots that cannot be factored by inspection, which is exactly when the quadratic formula becomes necessary.

What happens if a = 0?

The equation is no longer quadratic — it becomes linear (bx + c = 0), and the quadratic formula does not apply since dividing by 2a would mean dividing by zero.

What does a repeated root look like graphically?

When the discriminant is exactly zero, the parabola touches the x-axis at exactly one point rather than crossing it at two, meaning the vertex sits directly on the x-axis.

What do complex roots mean in practice?

Complex roots mean the parabola never crosses the x-axis, so there is no real-numbered solution — this often signals that a physical model (like height or profit) has no real-world zero in the range being considered.

How can I visualize the roots this calculator gives me?

Graphing y = ax² + bx + c and looking at where the curve crosses the x-axis shows the real roots visually, which is a good way to double-check the numeric roots from the formula.

The Quadratic Formula

A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. The quadratic formula x = (−b ± √D) / 2a always works, regardless of whether the equation can be factored. The discriminant D = b² − 4ac tells you the nature of roots before solving.

The vertex form of the parabola is y = a(x − h)² + k where h = −b/2a and k = f(h). The axis of symmetry is the vertical line x = h that the parabola is symmetric about.

Example

Solve: x² − 5x + 6 = 0 (a=1, b=−5, c=6)

1D = 25 − 24 = 1 (positive → 2 real roots)

2x = (5 ± 1) / 2

✓ x₁ = 3, x₂ = 2 (factors: (x−3)(x−2))

Frequently Asked Questions

What is the quadratic formula and when should I use it?

The quadratic formula solves ax² + bx + c = 0 for x: x = (−b ± √(b² − 4ac)) ÷ 2a. Use it when factoring is difficult or impossible. It always works — factoring only succeeds when the roots are rational, but the quadratic formula handles irrational and complex roots too.

What does the discriminant tell me?

The discriminant is b² − 4ac. If greater than 0: two distinct real roots. If equal to 0: one repeated real root (the parabola just touches the x-axis). If less than 0: two complex (imaginary) roots (the parabola doesn't cross the x-axis). The discriminant quickly tells you how many and what type of solutions to expect before solving.

What are real-world applications of quadratic equations?

Projectile motion (how high and how far an object travels), profit optimisation (finding the maximum profit from a quadratic profit function), area problems (dimensions that produce a given area), and circuit design (resonant frequency calculations). Quadratic equations appear wherever a relationship involves a squared variable.