Quadratic Equation Solver

Solve any quadratic equation with step-by-step solutions

Find roots, discriminant, vertex, and visualize parabolas with detailed analysis

Last updated: November 25, 2025

CreatorFrank Zhao

Formula form

Ax² +Bx +C= 0A(x -H)² +K= 0A(x -x₁)(x -x₂)= 0

A - Coefficient of x²

B - Coefficient of x

C - Constant term

Allow negative discriminant

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Introduction / Overview

This calculator solves quadratic equations and helps you understand them visually. You can work in standard form, vertex form, or factored form, and the solver will show the discriminant, roots, vertex, and a parabola plot.

🧠 Think of it as a “quick answer + explanation” tool: it gives the roots, but also tells you what kind of roots you have and why.

Who is this for?

Students checking homework, engineers validating a design curve, and anyone who wants to understand where a parabola crosses the x-axis.

Why the results are reliable

The solver uses the classic discriminant + quadratic formula approach and keeps forms consistent when you switch between them.

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Tip: If you need logarithms for related algebra steps (like solving exponentials), our Log Calculator is a handy companion.

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How to Use / Quick Start Guide

Pick the form you have

Most textbooks give $Ax^2 + Bx + C = 0$ . If you already know the vertex, switch to $A(x-H)^2 + K = 0$ .

Enter the parameters

Type the numbers exactly as you see them. The calculator updates automatically once the required inputs are filled.

Read the discriminant first

The discriminant $\Delta$ tells you if the roots are real or complex.

Use the result cards

You’ll see the roots, the function in different forms, and a plot. If you want the full derivation, enable “Show step by step solution”.

Example 1: Two real roots

Solve $2x^2 + 5x - 3 = 0$ .

Discriminant

\Delta = B^2 - 4AC

=

5^2 - 4\cdot 2\cdot (-3)

=

25 + 24

=

49

Since $\Delta > 0$ , there are two distinct real roots.

x = \frac{-B \pm \sqrt{\Delta}}{2A}

=

\frac{-5 \pm 7}{4}

=

\{-3,\ 0.5\}

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Interpretation: the parabola crosses the x-axis at $x=-3$ and $x=0.5$ .

Example 2: Complex roots (when Δ is negative)

If you enter $123x^2 + 2x + 3 = 0$ , the discriminant is negative.

\Delta = 2^2 - 4\cdot 123\cdot 3

=

4 - 1476

=

-1472

Enable “Allow negative discriminant” to see the complex solutions in the form $a \pm bi$ .

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Real-World Examples / Use Cases

1) Finding break-even points

If a model gives profit $P(x)=ax^2+bx+c$ , the break-even points are where $P(x)=0$ .

2) Vertex as an optimum

The vertex $(H,K)$ is a minimum or maximum (depending on the sign of $A$ ). Great for optimization problems.

3) Projectile motion timing

Vertical position often looks like $y(t)=at^2+bt+c$ . Solve $y(t)=0$ to find when something hits the ground.

4) Checking graph intersections

Intersections with the x-axis are exactly the roots. The plot helps you sanity-check the sign and rough size.

Worked scenario: vertex from coefficients

For $f(x)=2x^2+5x-3$ , the vertex x-coordinate is $x_v=-\frac{B}{2A}$ .

x_v=-\frac{5}{2\cdot 2}

=

-\frac{5}{4}

=

-1.25

Then $y_v=f(x_v)$ gives the maximum/minimum value.

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Common Scenarios / When to Use

This solver is especially useful when:

You need roots quickly (homework checks, test practice, quick verification).
You care about the vertex (optimization / maximum / minimum problems).
The discriminant is negative and you still want a clean complex-number answer.

⚠️ Not a quadratic? If $A=0$ , the equation becomes linear, not quadratic. In that case, solve $Bx+C=0$ instead.

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Tips & Best Practices

Keep units consistent

If your quadratic comes from physics or geometry, make sure the underlying units match before solving.

Use the plot as a sanity check

If the plot shows the parabola far above the x-axis, you should expect complex roots.

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Common mistake: forgetting the equation is set equal to $0$ . If you have something like $2x^2+5x-3=7$ , rewrite it as $2x^2+5x-10=0$ before entering values.

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Calculation Method / Formula Explanation

The solver uses a few core ideas: the discriminant, the quadratic formula, and the vertex relationship.

Discriminant

\Delta = B^2 - 4AC

$\Delta > 0$ : 2 real roots

$\Delta = 0$ : 1 repeated root

$\Delta < 0$ : complex roots

Quadratic formula

x = \frac{-B \pm \sqrt{\Delta}}{2A}

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Vertex

Standard form to vertex

x-coord

x_v = -\frac{B}{2A}

y-coord

y_v = f(x_v)

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Nice shortcut: In vertex form $A(x-H)^2 + K = 0$ , the discriminant simplifies to $\Delta = -4AK$ .

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Related Concepts / Background Info

A few concepts make quadratics feel much less mysterious.

Discriminant intuition

It’s a “crosses the x-axis?” detector. Positive means two crossings, zero means a touch, negative means no real crossings.

Complex roots

When $\Delta<0$ , the roots are still valid — they just live in the complex plane.

What the sign of A means

If $A>0$ , the parabola opens upward and the vertex is a minimum.
If $A<0$ , it opens downward and the vertex is a maximum.

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Frequently Asked Questions (FAQs)

Why does the calculator require A to be non-zero?

Because a quadratic needs a true $x^2$ term. If $A=0$ , the equation becomes linear.

What does a negative discriminant mean?

It means there are no real roots. The solutions are complex and can be written as $a \pm bi$ .

Why do I sometimes see approximations (≈)?

Some roots involve irrational numbers like $\sqrt{2}$ . The calculator shows a numeric approximation for quick use.

Can I use vertex form directly?

Yes. If you know $A$ , $H$ , and $K$ from $A(x-H)^2 + K = 0$ , you can enter them and solve immediately.

How can I quickly verify a root?

Plug it back into $Ax^2 + Bx + C$ and check if you get $0$ (within rounding tolerance).

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Limitations / Disclaimers & Sources

⚠️ This calculator is for educational and general problem-solving use. It doesn’t replace professional judgment in engineering, safety-critical design, or regulated work.

Practical limitations

Very large or very tiny coefficients can amplify rounding effects in floating-point arithmetic.
Displayed values may be rounded for readability; use the exact forms when precision matters.