📈 Quadratic Equation Solver
x² − 3x − 10 = 0 → x = 5 or x = −2, exactly — and when roots aren't rational you get the simplified radical form (3 ± √89)/2, not just a decimal.
Coefficients accept integers, decimals and fractions (e.g. 1/2). Negative b or c: type the minus sign.
two rational roots (D is a perfect square)
x = 5 , -2
Working
- Cleared to integer coefficients: 1x² − 3x − 10 = 0
- Discriminant D = b² − 4ac = (-3)² − 4·1·-10 = 49
- Quadratic formula: x = (−b ± √D) / 2a = (3 ± √49) / 2
- Simplified exactly to the roots above. Vertex of the parabola: (3/2, -49/4)
- Factored form: (x − 5)(x + 2) = 0
Roots come out exact — simplified fractions and radicals like (3 + √89)/2, not just decimals. Runs locally.
How the quadratic equation solver works
Coefficients (integers, decimals or fractions) are first cleared to integers, then the discriminant D = b² − 4ac decides the character of the roots: a perfect square gives two rational roots (shown as simplified fractions), a positive non-square gives irrational roots — which the tool presents in exact simplified surd form by extracting the largest square factor from D — a zero discriminant gives one repeated root, and a negative one gives a complex conjugate pair (a ± bi). Every case shows the numbered working: the integerized equation, the discriminant computation, the quadratic formula with your values, and the parabola's vertex.
The exact surd form is what most online solvers skip: x² − 3x − 20 = 0 has roots (3 ± √89)/2, and "5.216990…" is a shadow of that answer — unusable if the next line of your homework needs the exact value. Extracting square factors matters too: √72 should appear as 6√2. The discriminant is worth internalizing: it tells you how many real roots exist before you solve anything.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a. The expression under the root, D = b² − 4ac, is the discriminant — its sign determines whether the roots are two real numbers, one repeated, or a complex pair.
What does the discriminant tell me?
D > 0: two distinct real roots (rational if D is a perfect square). D = 0: one repeated root. D < 0: no real roots — a complex conjugate pair. The tool reports which case yours is.
Why show roots as radicals instead of decimals?
Because (3 + √89)/2 is the answer, while 6.2169… is an approximation of it. Exact form matters for algebra homework, further symbolic manipulation, and checking by substitution. Decimals are shown alongside for practical use.
Does it handle complex roots?
Yes — when D < 0 the roots appear as an exact conjugate pair like (1 ± √7·i)/2, with decimal approximations.
Can coefficients be fractions or decimals?
Yes — 0.5x² + x − 1/3 = 0 works: coefficients are cleared to integers exactly (multiply by the common denominator) before solving, so no precision is lost.