functions

Quadratic Formula Calculator | Find roots and discriminant

Q: How do I use the quadratic formula calculator?

Enter the coefficients a, b, and c from your equation written as ax² + bx + c = 0, then press calculate. The tool applies x = (−b ± √(b² − 4ac)) / 2a and returns both real roots, or tells you when the discriminant is negative and the roots are complex. The discriminant step is shown so you can follow the working.

Q: What does the discriminant tell me before solving?

Everything about the root type: b² − 4ac > 0 means two distinct real roots, = 0 means one repeated root, < 0 means a complex conjugate pair. For 2x² − 7x + 3 the discriminant is 25, so two real roots were guaranteed before the formula was even applied.

Q: Can it handle complex roots?

Yes — a negative discriminant yields a conjugate pair like 2 ± 3i, computed with the same formula; the square root of the negative part becomes the imaginary component. The page shows the full complex working, not just a ‘no real solution' message.

Q: Why do my results differ from another calculator?

The Quadratic Formula Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Enter the coefficients a, b, and c of a quadratic equation (ax2 + bx + c = 0) and this calculator solves it with the quadratic formula, returning both real roots when they exist. It computes the discriminant to show whether the roots are real, repeated, or There is no limit on usage, and it works on any device.

Q: When is factoring faster than the quadratic formula?

When the roots are nice integers you can spot — but the formula never fails and never needs guessing. It's also self-checking: the discriminant step reveals the root type upfront, so you know what kind of answer to expect before simplifying.

Q: What if a = 0?

Then it isn't a quadratic equation at all — it's linear (bx + c = 0) with the single root x = −c/b. The page detects this case and solves the linear equation instead of dividing by zero, which the quadratic formula would attempt.

Q: How do I check the roots by hand?

Substitute each root back into the equation — both must give zero. For 2x² − 7x + 3 = 0: x = 3 gives 18 − 21 + 3 = 0 ✓, and x = 0.5 gives 0.5 − 3.5 + 3 = 0 ✓. The page shows the substitution step so verification takes seconds.

Quick reference

Discriminant D	Root type
D > 0	Two real roots
D = 0	One repeated real root
D < 0	Two complex roots

functions Shows the working, not just the answer

For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

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Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

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Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

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Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

Coefficient	Meaning
a	Quadratic term; cannot be 0
b	Linear term
c	Constant term

lightbulb Solving 2x² − 7x + 3 = 0

Find where the parabola y = 2x² − 7x + 3 crosses the x-axis. Here a = 2, b = −7, c = 3. The discriminant is b² − 4ac = 49 − 24 = 25, and √25 = 5.

Result: x = (7 ± 5) ÷ 4, giving x = 12/4 = 3 and x = 2/4 = 0.5.

What this means: The parabola crosses the x-axis at 0.5 and 3 — verify by substitution: 2(3)² − 7(3) + 3 = 18 − 21 + 3 = 0 ✓, and 2(0.5)² − 7(0.5) + 3 = 0.5 − 3.5 + 3 = 0 ✓. A positive discriminant of 25 promised two real roots before any solving began.

Formula & methodology

Formula: x = (−b ± √(b² − 4ac)) / 2a

How the quadratic formula works

Any equation in the form ax² + bx + c = 0 is solved by x = (−b ± √(b² − 4ac)) ÷ 2a. The expression under the root — the discriminant — classifies the answer before solving: positive means two real roots, zero means one repeated root, negative means a complex conjugate pair.

Read the coefficients a, b, c (a ≠ 0 — otherwise the equation is linear).
Compute the discriminant b² − 4ac and note its sign.
Take its square root, apply ±, and divide by 2a.
Verify by substituting each root back into the equation.

Example: 2x² − 7x + 3 = 0 → discriminant 25 → x = (7 ± 5)/4 = 3 or 0.5. Complex roots like 2 ± 3i use the identical steps, with the imaginary unit carrying the negative part.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do I use the quadratic formula calculator?

Enter the coefficients a, b, and c from your equation written as ax² + bx + c = 0, then press calculate. The tool applies x = (−b ± √(b² − 4ac)) / 2a and returns both real roots, or tells you when the discriminant is negative and the roots are complex. The discriminant step is shown so you can follow the working.

What does the discriminant tell me before solving?

Everything about the root type: b² − 4ac > 0 means two distinct real roots, = 0 means one repeated root, < 0 means a complex conjugate pair. For 2x² − 7x + 3 the discriminant is 25, so two real roots were guaranteed before the formula was even applied.

Can it handle complex roots?

Yes — a negative discriminant yields a conjugate pair like 2 ± 3i, computed with the same formula; the square root of the negative part becomes the imaginary component. The page shows the full complex working, not just a ‘no real solution' message.

Why do my results differ from another calculator?

The Quadratic Formula Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Enter the coefficients a, b, and c of a quadratic equation (ax2 + bx + c = 0) and this calculator solves it with the quadratic formula, returning both real roots when they exist. It computes the discriminant to show whether the roots are real, repeated, or There is no limit on usage, and it works on any device.

When is factoring faster than the quadratic formula?

When the roots are nice integers you can spot — but the formula never fails and never needs guessing. It's also self-checking: the discriminant step reveals the root type upfront, so you know what kind of answer to expect before simplifying.

What if a = 0?

Then it isn't a quadratic equation at all — it's linear (bx + c = 0) with the single root x = −c/b. The page detects this case and solves the linear equation instead of dividing by zero, which the quadratic formula would attempt.

How do I check the roots by hand?

Substitute each root back into the equation — both must give zero. For 2x² − 7x + 3 = 0: x = 3 gives 18 − 21 + 3 = 0 ✓, and x = 0.5 gives 0.5 − 3.5 + 3 = 0 ✓. The page shows the substitution step so verification takes seconds.

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