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Quadratic Formula Calculator

Solve quadratic equations of the form ax² + bx + c = 0.

ax² + bx + c = 0

Quadratic equation formulas

Quadratic formulax = −b ± √b2 − 4ac2a
DiscriminantΔ = b2 − 4ac
Vertexx = −b2a
Sum and product of rootsx1 + x2 = −ba, x1x2 = ca

Where For ax2 + bx + c = 0 with a ≠ 0. Δ > 0: two real roots · Δ = 0: one repeated root · Δ < 0: two complex roots

Worked examples

  1. Solve x² − 5x + 6 = 0.

    1. Δ = 25 − 24 = 1
    2. x = 5 ± 12

    Answer: x = 3 or x = 2

  2. Solve 2x² + 4x + 5 = 0.

    1. Δ = 16 − 40 = −24 (negative → complex roots)
    2. x = −4 ± √−244 = −4 ± 2i√64

    Answer: x = −1 ± 1.2247i

  3. Solve x² − 6x + 9 = 0.

    1. Δ = 36 − 36 = 0
    2. x = 62

    Answer: x = 3 (double root)

Tips & common mistakes

  • Try factoring first — x² − 5x + 6 = (x − 2)(x − 3) is faster than the formula.
  • Divide every term by a common factor before using the formula to keep numbers small.