How does the discriminant tell me the kind of roots?

The discriminant Δ = b² − 4ac is what's under the square root in the quadratic formula. If Δ > 0 the square root is a real number and you get two distinct real roots. If Δ = 0 the square root is zero so both ± branches collapse into one repeated real root. If Δ < 0 the square root pulls an imaginary unit i out, and you get a conjugate pair p ± qi. You don't have to factor anything to find out which case you're in — just plug a, b, c in and read Δ.

Why are the complex roots always conjugates?

Because a, b, c are real numbers. The quadratic formula evaluates to (−b/2a) ± √Δ/(2a). When Δ < 0, √Δ = i·√|Δ| pulls the i out as a pure imaginary number, and the ± gives two roots with the same real part and equal-but-opposite imaginary parts — that's the definition of a complex conjugate pair. It's a property of any polynomial with real coefficients, not a quirk of the quadratic.

What does the vertex tell me about the parabola?

The vertex (−b/2a, (4ac−b²)/4a) is the parabola's turning point. If a > 0 the parabola opens upward and the vertex is the minimum; if a 0 the vertex is on the opposite side of zero from the parabola's open mouth.

How do you compute roots when a is very large or very small?

The naive (−b ± √Δ)/(2a) formula can lose precision when b and √Δ are nearly equal in magnitude (one of the two roots will subtract two close numbers and lose its leading digits). This solver uses the numerically stable variant — pick the sign of −b in the numerator so the two terms add rather than cancel, then recover the second root by Vieta's product law c/(a·r₁). That keeps the error to within a few ulps even for stiff coefficients.

Can I enter coefficients as fractions like 1/2?

Yes. The input parser accepts integers (3), decimals (0.5), fractions (1/2, −3/4), and negatives. Internally the fraction is divided to a JavaScript number and the solver works as normal. If your fraction is something irrational that you wrote as a long decimal (like 0.33333), the result will reflect that truncation; for exact rationals just type the fraction.

Does completing the square give the same answer as the quadratic formula?

Yes — they're algebraically identical. Completing the square is how you derive the quadratic formula. The solver shows both views so you can see (a) the mechanical plug-and-chug version (the formula) and (b) the geometric version that explains why the vertex is at −b/2a. Vieta's check (sum = −b/a, product = c/a) is a third view that doesn't find the roots but does verify them once you have them.

Why doesn't the parabola cross the x-axis when Δ < 0?

The x-intercepts of y = ax² + bx + c are exactly where y = 0, i.e. the real solutions of ax² + bx + c = 0. When Δ 0) or below (a < 0) it.

Verify a homework answer before turning it in

You factored x² − 5x + 6 by hand and got x = 2, 3 — but you're not 100% sure of the signs. Plug a=1, b=−5, c=6 into the solver. It returns Δ = 1, two distinct real roots 2 and 3, vertex at (2.5, −0.25). The discriminant being a perfect square confirms the equation factors over the integers, and the three-method panel shows the quadratic formula, completing the square, and a Vieta's check (sum = 5, product = 6) all in one screen.

Understand why a parabola "doesn't have roots"

A textbook problem asks you to solve x² + 2x + 5 = 0 and the answer key says "no real solutions". Plug it in: Δ = −16, the solver shows the complex conjugate pair −1 + 2i and −1 − 2i, and the parabola visualisation sits entirely above the x-axis with its minimum at (−1, 4). You can see, not just be told, why the curve never touches y = 0.

Find the vertex of a projectile motion problem

Physics homework: a ball's height is h(t) = −5t² + 20t + 1. What's the maximum height and when does it happen? Enter a=−5, b=20, c=1. The vertex is (2, 21) — the ball reaches 21 metres at t = 2 seconds. The parabola opens downward (a < 0) as expected, and you don't need calculus to derive it; the vertex formula does the work.

Compare three solution methods side by side when learning

Students typically meet the quadratic formula first and never quite understand where −b/2a comes from. Enter any equation and read the three method panels: the formula gives the mechanical answer, completing-the-square explains the geometry (the vertex jumps out as −b/2a), and Vieta's formulas show the sum-and-product structure that becomes the basis of polynomial root theory in later years.

Spot whether an equation has integer roots

Before reaching for the formula, students often try to factor. The discriminant tells you instantly whether factoring will give nice numbers: Δ a perfect square ⇒ rational roots, otherwise you'll need the formula or completing the square. The solver returns Δ directly, so you can decide your strategy in one second before committing to a method.

Quadratic Equation Solver — Discriminant, Roots, Vertex, Parabola

Quadratic equation solver — ax² + bx + c = 0, discriminant analysis, real + complex roots, vertex/intercepts, parabola visualization.

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

ax² + bx + c = 0

a (quadratic)

b (linear)

c (constant)

Accepts integers, decimals (0.5), fractions (1/2), negatives (−3).

Try a textbook example

Discriminant Δ = b² − 4ac

Δ = 1

Two distinct real roots (Δ > 0)

Roots

x₁ = 2

x₂ = 3

Vertex

(2.5, -0.25)

Axis of symmetry

x = 2.5

y-intercept

(0, 6)

x-intercepts

(2, 0), (3, 0)

Opens

upward (a > 0)

Parabola

Vertex marked ◆, real roots marked ●, y-intercept marked ◌.

Three ways to see the solution

Quadratic formula

Plug a=1, b=-5, c=6 into x = (−b ± √Δ) / 2a. With Δ=1, the roots are x = 2, 3.

Completing the square

Rewrite as 1(x − h)² + k = 0 with h=2.5, k=-0.25. Then x = h ± √(−k/a).

Vieta's formulas (verify)

For ax² + bx + c = 0, the sum of roots = −b/a = 5 and product = c/a = 6. Check this against the roots above.

10 classic exercises (with answers)

Click any row to load it into the solver and see the full breakdown.

All math runs locally — your numbers never leave this tab.

What this tool does

A complete solver for ax² + bx + c = 0 that returns every number a student or teacher actually needs in one pass, not just "the answer". Plug in any a, b, c (accepts integers, decimals like 0.5, fractions like 1/2, and negatives) and the page returns: the discriminant Δ = b² − 4ac, the kind of solution (two distinct real roots, one repeated real root, or a conjugate complex pair with the i), the vertex at (−b/2a, (4ac−b²)/4a), the axis of symmetry x = −b/2a, the y-intercept (0, c), the x-intercepts (which coincide with the real roots when they exist), and the opening direction. The solution is shown three different ways side by side: the quadratic formula, completing the square, and Vieta's formulas (so you can cross-check sum and product of roots). A dynamic SVG draws the parabola with axis labelled — the viewport auto-fits the vertex and roots so the curve always frames the interesting region. Ten classic textbook exercises with answers are included and one-click load into the solver. All math runs in your browser; no input is ever uploaded.

Tool details

Input: Text + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy + Preview; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Local preference storage; Preferences, history, or drafts are saved in this browser without an account.
Performance budget: Initial JS <= 18 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Quadratic Equation Solver fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Verify a homework answer before turning it in
You factored x² − 5x + 6 by hand and got x = 2, 3 — but you're not 100% sure of the signs. Plug a=1, b=−5, c=6 into the solver. It returns Δ = 1, two distinct real roots 2 and 3, vertex at (2.5, −0.25). The discriminant being a perfect square confirms the equation factors over the integers, and the three-method panel shows the quadratic formula, completing the square, and a Vieta's check (sum = 5, product = 6) all in one screen.
Understand why a parabola "doesn't have roots"
A textbook problem asks you to solve x² + 2x + 5 = 0 and the answer key says "no real solutions". Plug it in: Δ = −16, the solver shows the complex conjugate pair −1 + 2i and −1 − 2i, and the parabola visualisation sits entirely above the x-axis with its minimum at (−1, 4). You can see, not just be told, why the curve never touches y = 0.
Find the vertex of a projectile motion problem
Physics homework: a ball's height is h(t) = −5t² + 20t + 1. What's the maximum height and when does it happen? Enter a=−5, b=20, c=1. The vertex is (2, 21) — the ball reaches 21 metres at t = 2 seconds. The parabola opens downward (a < 0) as expected, and you don't need calculus to derive it; the vertex formula does the work.
Compare three solution methods side by side when learning
Students typically meet the quadratic formula first and never quite understand where −b/2a comes from. Enter any equation and read the three method panels: the formula gives the mechanical answer, completing-the-square explains the geometry (the vertex jumps out as −b/2a), and Vieta's formulas show the sum-and-product structure that becomes the basis of polynomial root theory in later years.
Spot whether an equation has integer roots
Before reaching for the formula, students often try to factor. The discriminant tells you instantly whether factoring will give nice numbers: Δ a perfect square ⇒ rational roots, otherwise you'll need the formula or completing the square. The solver returns Δ directly, so you can decide your strategy in one second before committing to a method.

Common pitfalls

Treating a = 0 as a valid input. If a = 0 the equation is no longer quadratic; the solver returns an explicit error rather than silently dividing by zero. Pull the linear case (bx + c = 0) into a separate calculation.
Confusing the discriminant's sign with the parabola's opening direction. Δ tells you how many real roots, a's sign tells you which way it opens — they are independent. A downward parabola can still have two real roots, and an upward one can have none.
Forgetting the ± in the quadratic formula. Both signs of √Δ are roots; if you only carry one, you're missing the other. The solver always lists both x₁ and x₂ even when they coincide, so the structure stays explicit.

Privacy

a, b, c and every derived number (discriminant, roots, vertex, intercepts) are computed locally in JavaScript. No coefficient you type and no result is ever sent to a server, logged, or analyzed. The solver works offline once the page is loaded — open it on a phone in airplane mode and it still works.

FAQ

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Quadratic Equation Solver — Discriminant, Roots, Vertex, Parabola

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Quadratic Equation Solver fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Verify a homework answer before turning it in

Understand why a parabola "doesn't have roots"

Find the vertex of a projectile motion problem

Compare three solution methods side by side when learning

Spot whether an equation has integer roots

Common pitfalls

Privacy

FAQ

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