Pythagorist
Algebra

Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula — with full step-by-step working. Enter the values of a, b and c to find the roots instantly. The calculator shows the discriminant, square root and both solutions clearly.

Solving: 1x² -5x +6 = 0

1x² -5x +6 = 0

x₁

3

x₂

2

How it works

  1. Write the equation in standard form

    1x2−5x+6=01x^2 -5x +6 = 0

    The equation is already in ax² + bx + c = 0 form. Here a = 1, b = -5, c = 6.

  2. Write the quadratic formula

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

    For any quadratic ax² + bx + c = 0, the solutions are given by the quadratic formula.

  3. Calculate the discriminant

    b2−4ac=−52−4×1×6=25−24=1b^2 - 4ac = -5^2 - 4 \times 1 \times 6 = 25 - 24 = 1

    The discriminant b² − 4ac = 1. Since it is positive, there are two distinct real roots.

    Result:Discriminant = 1

  4. Calculate the square root of the discriminant

    1=1\sqrt{1} = 1

    1 is a perfect square. √1 = 1.

    Result:√1 = 1

  5. Calculate x₁ (using +)

    x1=−−5+12=62=3x_1 = \frac{--5 + 1}{2} = \frac{6}{2} = 3

    Using the + sign: x₁ = (−-5 + 1) ÷ 2 = 3.

    Result:x₁ = 3

  6. Calculate x₂ (using −)

    x2=−−5−12=42=2x_2 = \frac{--5 - 1}{2} = \frac{4}{2} = 2

    Using the − sign: x₂ = (−-5 − 1) ÷ 2 = 2.

    Result:x₂ = 2

The quadratic formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
aaCoefficient of x² (must not be 0)
bbCoefficient of x
ccConstant term
b2−4acb² − 4acThe discriminant — determines number of roots

Example substitution

x2−5x+6=0⇒x=5±25−242=5±12x^2 - 5x + 6 = 0 \Rightarrow x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}

a=1, b=−5, c=6. Discriminant = 25−24 = 1. Roots: x = (5+1)/2 = 3 and x = (5−1)/2 = 2.

Worked examples

Solve x² − 5x + 6 = 0

  1. 1
    a=1, b=−5, c=6:
  2. 2
    Discriminant: (−5)² − 4(1)(6) = 25 − 24 = 1
  3. 3
    x₁: (5 + 1)/2 = 3
  4. 4
    x₂: (5 − 1)/2 = 2
Answer: x = 3 or x = 2

Solve 2x² + 5x − 3 = 0

  1. 1
    a=2, b=5, c=−3:
  2. 2
    Discriminant: 25 − 4(2)(−3) = 25 + 24 = 49
  3. 3
    √49: = 7
  4. 4
    x₁: (−5+7)/4 = 0.5
  5. 5
    x₂: (−5−7)/4 = −3
Answer: x = 0.5 or x = −3

Solve x² + 4x + 4 = 0

  1. 1
    Discriminant: 16 − 4(1)(4) = 0
  2. 2
    One root: x = −4/2 = −2
Answer: x = −2 (repeated root)

Frequently asked questions

What is the quadratic formula?+

For ax² + bx + c = 0, the solutions are x = (−b ± √(b²−4ac)) / 2a.

What is the discriminant?+

The discriminant is b² − 4ac. If positive: two real roots. If zero: one repeated root. If negative: no real roots.

How do I use the quadratic formula step by step?+

Identify a, b, c. Calculate the discriminant. Take the square root. Apply ± to get two answers. Divide by 2a.

What does "no real roots" mean?+

When the discriminant is negative, the quadratic has no real solutions — the parabola does not cross the x-axis.

When should I use the quadratic formula?+

Use it when the quadratic cannot be factorised easily, or when you need exact decimal answers.

How do I check my quadratic answer?+

Substitute each root back into ax² + bx + c — the result should equal zero.

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