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Quadratic Equation Calculator

Solve ax² + bx + c = 0 and see every step. Get real or complex roots as exact answers and decimals, plus the factored and vertex forms.

Level: Grades 9–11

  • Common Core standard HSA.REI.B.4.bSolve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
  • Common Core standard HSN.CN.C.7Solve quadratic equations with real coefficients that have complex solutions.
Printable worksheet

Quadratic Equation Calculator

Inputs

If a is 0, there's no x² term, so it's solved as the straight-line equation bx + c = 0.

Result

Roots

x = 3 or x = −1

x = 3 or x = −1. Discriminant 64; vertex at (1, −8).

Root x₁
3
Root x₂
-1
Factored form
2(x − 3)(x + 1)
Discriminant (b² − 4ac)
64
Type of roots
Two distinct real roots (D > 0)
Vertex (h, k)
(1, −8)
Vertex form
y = 2(x − 1)² − 8
Axis of symmetry
x = 1
y-intercept
(0, −6)

See the step-by-step solution ↓

Points on the graph of y = 2x² − 4x − 6
Points on the graph of y = 2x² − 4x − 6
xy
-210
-10
0-6
1-8
2-6
30
410

Step-by-step solution

  1. Start with your equation set equal to zero, where a = 2, b = −4 and c = −6

    2x² − 4x − 6 = 0

  2. The quadratic formula finds the roots: the values of x that make the equation true

    x=−b±D2a

  3. Work out D = b² − 4ac, called the discriminant. It sits under the square root and tells you how many real roots there are

    D=(−4)2−4×2×(−6)=16−(−48)=64

  4. D is positive, so there are two real roots. Put your numbers into the formula

    x=4±642×2=4±84

  5. The ± means "plus or minus". Use the + sign for one root

    4+84=3

  6. Then use the − sign for the other root

    4−84=−1

  7. So the two roots are

    x = 3 or x = −1

  8. Find the vertex, the curve's turning point. It sits on the line x = −b ÷ 2a, called h, which splits the curve into mirror halves. The curve opens upward, so the vertex is its lowest point

    h=42×2=1

  9. Find the vertex's height, k, by putting h back into the equation. That simplifies to k = c − b² ÷ 4a

    k=−6−(−4)24×2=−8

  10. Because the roots are whole numbers or fractions, the left side can be split into factors. Each root r gives a factor (x − r), scaled so the numbers match

    2(x − 3)(x + 1) = 0

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

How to solve this on a calculator

The keys to press to get this page's worked-example answer on an ordinary calculator, with what the display should show after each step. A named-model tab follows that calculator's official guidebook.

Both roots of 2x² − 4x − 6 = 0 with the quadratic formula

Problem
Solve 2x² − 4x − 6 = 0.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a
Example values
a = 2 (coefficient of x²); b = −4 (coefficient of x); c = −6 (constant term). These are the fixed values from the worked example on this page, so the keys stay the same whatever you type into the calculator above.
Why a scientific calculator
The formula needs squares, a square root and brackets, and Ans saves retyping the discriminant. It also works on a statistics calculator (not on this site yet) or graphing calculator.
The whole sequence
(−4)x²−4×2×(−6)=(4+√Ans))÷(2×2)=(4−√64))÷(2×2)=

Step by step

  1. (−4)x²

    Start with the discriminant, b² − 4ac. Type b in brackets, (−4), and press x², so the minus sign is squared too.

    Display: (−4)^2

  2. −4×2×(−6)

    Subtract 4ac: − 4 × 2 × (−6), with c in brackets.

    Display: (−4)^2−4×2×(−6)

  3. =

    Press =. The discriminant is 64. It's positive, so there are two real roots.

    Display: 64

  4. (4+√Ans))

    First root, with +. −b is 4, so type (4 + √Ans). Ans holds the discriminant, 64.

    Display: (4+√(ans))

  5. ÷(2×2)

    Divide by 2a, which is 2 × 2. Bracket it so the whole bottom is used.

    Display: (4+√(ans))÷(2×2)

  6. =

    Press =. One root is x = 3.

    Display: 3

  7. (4−√64))÷(2×2)

    Second root, with −. Ans now holds 3, so type the discriminant itself: (4 − √64) ÷ (2 × 2).

    Display: (4−√(64))÷(2×2)

  8. =

    Press =. The other root is x = −1.

    Display: -1

If your answer looks wrong

  • −4 x² without brackets gives −16, not 16: the square is done before the minus sign. Always bracket a negative b.
  • Leaving the brackets off 2 × 2 divides by 2 and then multiplies by 2, giving 12 instead of 3.
  • If √ shows an error, the discriminant is negative: there are no real roots, and the calculator above shows the complex ones.

How this is checked: an automated test presses these keys on this site's scientific calculator, checks every display along the way, and confirms the answer matches the calculator on this page (last checked 2026-10-19). It has not been checked on any particular brand or model, and real calculators label some keys differently.

What this calculator does

Solve any equation of the form ax² + bx + c = 0, and see every step so you can follow along. That's a quadratic equation, and a can't be zero. Its graph is a U-shaped curve called a parabola. The solutions, called roots, are the values of x where that curve crosses the x-axis. Every quadratic has exactly two roots, if you count a root that repeats and complex numbers (numbers that use i, the square root of −1). The quadratic formula finds them all.

Enter the three numbers a, b and c, and the calculator works through the formula with them. It shows the discriminant (the part under the square root), each sign of the ± and the roots as decimals. For whole numbers, it also gives the exact roots the way a teacher expects, such as x = 1 ± √2 or x = (−1 ± i√3)/2. When the roots are whole numbers or fractions, it gives the factored form too, such as 2(x − 3)(x + 1). It also finds the vertex, or turning point, and the vertex form. You get the line the curve is mirrored across and where it crosses the y-axis. A list of points helps you sketch the curve. If a is zero, the equation isn't quadratic anymore, so the calculator solves it as a straight-line equation instead.

Where it's used

Why this math matters outside the classroom:

  • Projectile flight. A ball thrown upward at 20 m/s from 1.5 m lands when −4.9t² + 20t + 1.5 = 0, about 4.2 seconds later.
  • Garden layout. A gardener with 20 m of edging who wants a 24 m² bed solves x(10 − x) = 24, so the sides are 4 m and 6 m.
  • Break-even prices. A shop whose profit is −2p² + 60p − 250 dollars finds its break-even prices of $5 and $25 by setting profit to zero.
  • Arch and bridge design. Designers find where the arch y = −0.02x² + 18 meets the ground by solving −0.02x² + 18 = 0, giving a 60 m span.
  • Electrical circuits. Engineers solve a quadratic for a resistor, inductor and capacitor circuit, and complex roots tell them the current oscillates as it fades.

Grade level and standards: Grades 9–11; Common Core State Standards for Mathematics HSA.REI.B.4.b, HSN.CN.C.7. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Rearrange your equation so one side is zero. Then read off a (the number in front of x²), b (the number in front of x) and c (the number on its own). If a term is missing, its number is 0.
  2. Enter a, b and c. Negative numbers and decimals are fine.
  3. Read the roots at the top. Real roots are also listed as separate numbers. Complex roots are written as p ± qi. When the exact form differs from the decimals, it's listed as Exact roots.
  4. Check the factored form, discriminant, vertex and vertex form below. Use the table of points to sketch the graph. Open the working to see the formula used with your numbers.

What the inputs mean

a (the number in front of x²)
The number in front of x². If it's 0, the x² term disappears and the equation becomes the straight-line equation bx + c = 0.
b (the number in front of x)
The number in front of x. Enter it with its sign: for x² − 5x + 6, b is −5.
c (the number on its own)
The number on its own, again with its sign.

Formula and method

D = b² − 4ac
x = (−b ± √D) ÷ 2a
vertex: h = −b ÷ 2a,  k = c − b² ÷ 4a
vertex form: y = a(x − h)² + k
factored form: a(x − x₁)(x − x₂)

In the formulas, a, b and c are your three numbers, and x is the unknown you're solving for. D is the discriminant. The vertex, or turning point, is at (h, k): h is how far across and k is how high. x₁ and x₂ are the two roots.

The discriminant D tells you what kind of answer to expect. When D is positive, there are two different real roots. When it's zero, the two roots are the same. When it's negative, its square root is imaginary, meaning it uses i, the square root of −1. Then the roots are a matching complex pair, −b ÷ 2a ± (√|D| ÷ 2|a|)i. Here |D| means D without its minus sign.

The vertex is halfway between the roots, on the line x = h that splits the curve into mirror halves. For two real roots, the calculator uses a rearranged version of the formula that gives the same answer. It avoids losing digits when b² is far bigger than 4ac.

For the exact form, √D is simplified by taking out its largest square factor, so √8 = 2√2. Then any factor shared by −b, the number in front of the root and 2a is divided out. Decimals are first multiplied by a power of ten to make them whole numbers, which doesn't change the roots. When D is a perfect square, the roots are whole numbers or fractions, and the equation splits into factors with whole numbers.

Worked example

Solve 2x² − 4x − 6 = 0.

  1. Here a = 2, b = −4 and c = −6.
  2. Discriminant: D = (−4)² − 4 × 2 × (−6) = 16 + 48 = 64, which is positive, so there are two real roots.
  3. x = (4 ± √64) ÷ 4 = (4 ± 8) ÷ 4, giving x = 12 ÷ 4 = 3 and x = −4 ÷ 4 = −1.
  4. Taking the + sign gives (4 + 8) ÷ 4 = 3; taking the − sign gives (4 − 8) ÷ 4 = −1.
  5. Both roots are whole numbers, so the quadratic factors as 2(x − 3)(x + 1).
  6. Vertex: h = 4 ÷ 4 = 1 and k = −6 − 16 ÷ 8 = −8, so the vertex form is y = 2(x − 1)² − 8.

Result: The roots are x = 3 and x = −1, and the factored form is 2(x − 3)(x + 1). The curve's lowest point is the vertex (1, −8). It crosses the y-axis at (0, −6).

Understanding the result

Real roots are the points where the graph crosses the x-axis. A double root means the curve just touches the axis at its vertex. Complex roots mean the curve never reaches the axis at all.

If a is positive, the curve opens upward and the vertex is its lowest point. If a is negative, it opens downward and the vertex is its highest point.

Assumptions and limitations

  • Your numbers a, b and c are real numbers.
  • The math uses the computer's standard decimal precision, and answers are rounded for display. A discriminant that is zero apart from tiny rounding errors is treated as exactly zero.
  • It doesn't solve equations whose numbers are complex, or equations with x³ or higher powers.
  • Exact roots and the factored form are given only for whole numbers, or decimals with up to six places. They also need to be no larger than 1,000,000 once the decimals are cleared. Other inputs get decimal answers only.
  • The vertex and vertex form are shown as decimals, not fractions.

Frequently asked questions

What does the discriminant tell me?

It tells you how many real solutions there are before you finish solving. Positive means two real roots. Zero means one root that repeats. Negative means no real roots, only a complex pair.

What happens if a is 0?

The equation becomes bx + c = 0, which has one solution, x = −c ÷ b. If b is also 0, there's nothing left to solve. The equation is either true for every x (when c is 0) or never true, and the calculator tells you which.

How should I read a complex answer like 1.5 ± 2.1i?

It stands for two roots: 1.5 + 2.1i and 1.5 − 2.1i, where i is the square root of −1. The real part, 1.5, is also how far across the vertex is.

Is factoring or completing the square more accurate?

All three methods, including the formula, give the same exact roots. The formula always works. Factoring by spotting the answer is only practical when the roots are tidy whole numbers or fractions.

How do I know if a quadratic can be factored?

Look at the discriminant. When a, b and c are whole numbers, the equation splits into brackets with whole numbers exactly when b² − 4ac is a perfect square, such as 0, 1, 4, 9 or 16. For 6x² + x − 2, D = 1 + 48 = 49 = 7². So it factors: (2x − 1)(3x + 2).

How do I write the answer in simplest radical form?

First simplify the square root of the discriminant. Then divide out any shared factor. For x² − 2x − 1 = 0, D = 8 and √8 = 2√2, so x = (2 ± 2√2) ÷ 2. Every term shares a factor of 2, which leaves x = 1 ± √2.

How do I convert to vertex form?

Find h = −b ÷ 2a. Then find k by putting h back into the equation. Write y = a(x − h)² + k. For 2x² − 4x − 6, h = 1 and k = −8, giving y = 2(x − 1)² − 8. The vertex form shows the turning point (h, k) at a glance.

How this calculator is checked

Method: Quadratic formula derived by completing the square; numerically stable root form q = −(b + sign(b)√D) ÷ 2; exact forms by simplifying √D over the integers. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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