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Quadratic Formula Calculator

Enter a, b and c from ax² + bx + c = 0 to find both roots, their exact simplified form, the discriminant and the vertex of the parabola.

Solves ax² + bx + c = 0. Use negative numbers for minus signs.

Solutions

x = 2, 1

Two real roots of x² − 3x + 2 = 0

x₁
2
x₂
1
Exact form
2, 1
Discriminant (b² − 4ac)
1
Vertex
(1.5, -0.25)
Axis of symmetry
x = 1.5

Step-by-step solution

Worked out with your numbers. Change any value above and the steps update.

  1. 1

    Identify a, b and c in ax² + bx + c = 0

    a = 1, b = -3, c = 2

  2. 2

    Discriminant: D = b² − 4ac

    (-3)² − 4 × 1 × 2 = 1

  3. 3

    D > 0, so there are two real roots. Square root of D

    √1 = 1

  4. 4

    Exact form

    x = (−b ± √D) ÷ 2a, simplified = 2, 1

  5. 5

    x₁ = (−b + √D) ÷ 2a

    (3 + 1) ÷ 2 = 2

  6. 6

    x₂ = (−b − √D) ÷ 2a

    (3 − 1) ÷ 2 = 1

How it works

  • The quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a.
  • The discriminant D = b² − 4ac tells you the type of answer: D > 0 gives two real roots, D = 0 one repeated root, and D < 0 two complex roots.
  • When a, b and c are whole numbers, the roots are also shown exactly, for example (2 ± √10)/3, with the square root and fraction simplified.
  • The vertex of the parabola is at x = −b ÷ 2a, which is also the axis of symmetry.

Worked examples

x² − 5x + 6 = 0

Solve x² − 5x + 6 = 0 with the quadratic formula.

  1. a = 1, b = −5, c = 6.
  2. D = (−5)² − 4 × 1 × 6 = 25 − 24 = 1, so √D = 1.
  3. x = (5 ± 1) ÷ 2.

Answer: x = 3 or x = 2

Try this example

A quadratic with no real roots

Solve 2x² + 4x + 5 = 0.

  1. D = 4² − 4 × 2 × 5 = 16 − 40 = −24, which is negative, so the roots are complex.
  2. √−24 = i√24 = 2i√6.
  3. x = (−4 ± 2i√6) ÷ 4 = (−2 ± i√6) ÷ 2.

Answer: x = (−2 ± i√6)/2 ≈ −1 ± 1.2247i

Try this example

Frequently asked questions

What if a is 0?

Then the equation is linear (bx + c = 0), not quadratic, and its single solution is x = −c ÷ b.

What does a negative discriminant mean?

There are no real solutions: the parabola never crosses the x-axis. The two roots are complex numbers of the form p ± qi.

How do I enter x² − 5 = 0?

Use a = 1, b = 0 and c = −5. Any missing term has a coefficient of 0.

Results are estimates for general information and are not financial, medical or professional advice.