JLJosh Liljenquist

Algebra calculator

Quadratic formula calculator

Solve a quadratic equation from coefficients a, b, and c. The calculator shows the discriminant first, then reports two real roots, one repeated root, or a complex-conjugate pair without attempting division when a is zero.

By

DiscriminantReal and complex rootsSubstitution shown

Enter known values

Enter the known values and select Calculate.

How to use this calculator

  1. Rewrite the equation in standard form ax² + bx + c = 0.
  2. Enter zero for a missing b or c term.
  3. Enter a nonzero value for a.
  4. Read the discriminant and root type before reviewing the calculated roots.

Formula and method

For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant D = b² − 4ac determines the root type.

Worked example

For x² − 5x + 6 = 0, D = 25 − 24 = 1. The roots are (5 + 1) ÷ 2 = 3 and (5 − 1) ÷ 2 = 2.

Assumptions and result interpretation

  • Coefficients are real finite numbers.
  • Complex roots are displayed in a + bi form when the discriminant is negative.
  • Decimal results are rounded for display while the formula uses full numeric precision.
  • The tool solves the equation; it does not factor or graph the polynomial.

Understanding the result

A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives two complex-conjugate roots.

Common mistakes

  • Leaving a missing coefficient blank instead of entering zero
  • Forgetting to move every term to one side
  • Dropping the ± sign
  • Dividing only the square-root term by 2a

Methodology sources

Right triangle calculatorStandard deviation calculator

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Use this result as a calculation aid. Confirm input definitions, required precision, and any governing policy or professional standard separately.