Normalize the equation first
The coefficient fields represent ax² + bx + c = 0. Move all terms to one side before entering a, b and c, and retain negative signs. Coefficient a must be nonzero.
If an equation is linear because a = 0, use the linear solver instead. Leaving a missing linear or constant term blank is also different from explicitly entering zero.
Connect root type with the discriminant
A positive discriminant produces two distinct real roots, a zero discriminant produces a repeated real root, and a negative one produces a complex pair. The output is a numerical approximation rather than an exact radical simplifier.
Check the results in the original polynomial. Near repeated roots, floating-point rounding can affect very small discriminants; the implementation uses relative numerical checks and rejects unusable ranges.
Use the root relationships as a second check
For ax² + bx + c = 0, the roots have sum −b/a and product c/a, counting a repeated root twice. For the worked example below, 3 + (−0.5) = 2.5 and 3 × (−0.5) = −1.5, matching −b/a and c/a. These relationships can reveal a missing minus sign or an incorrect denominator.
LET’S WORK IT THROUGH
Solve 2x² − 5x − 3 = 0.
- a = 2, b = −5, c = −3.
- D = 25 + 24 = 49.
- x = (5 ± 7)/4.
THE ANSWERx = 3 or x = −0.5.
YOUR TURN
A small check for understanding.
What happens to x² − 6x + 9 = 0?
Reveal the answer
Its discriminant is zero, so the repeated root is x = 3.
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.