Write zero on one side first
The formula applies to ax² + bx + c = 0, with a ≠ 0. Move every term to one side before identifying coefficients, and include zero for a missing term.
For x² = 5x − 6, the standard form is x² − 5x + 6 = 0. Its b value is −5, not 5. Carry the sign with the coefficient.
Read the discriminant
The discriminant D = b² − 4ac determines the root type. A positive D gives two distinct real roots; zero gives a repeated real root; a negative D gives a complex conjugate pair.
The ± symbol represents two calculations. Evaluate both the plus and minus branches, with the whole numerator divided by 2a. Parentheses prevent calculator-entry mistakes.
Check and interpret the roots
Substitution should make the original polynomial zero, allowing for numerical rounding. The sum of the roots is −b/a and the product is c/a, which provide additional checks.
The calculator returns approximate numerical roots, including complex roots. Very different coefficient magnitudes can exceed reliable floating-point evaluation and are rejected rather than shown as a misleading answer.
LET’S WORK IT THROUGH
Solve x² − 7x + 12 = 0.
- a = 1, b = −7, c = 12.
- D = 49 − 48 = 1.
- x = (7 ± 1)/2 gives 4 and 3.
THE ANSWERThe roots are 3 and 4.
YOUR TURN
A small check for understanding.
What type of roots does x² + 4 = 0 have?
Reveal the answer
Two complex roots, +2i and −2i, because the discriminant is −16.
Now try your own numbers.
The quadratic equation solver includes input guidance and the limits of its supported calculations. Use it alongside the method above.
Open Quadratic equation solver ↗A note on precision: preserve exact fractions where possible and round numerical results at the end. Found an unclear step? Prepare a correction report.