Type the coefficients, including fractions or decimals, and follow the full working: the discriminant, simplifying the square root, exact answers like (3 ± √5)/2 and their decimal values.
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Each line of the working is written out like a worked example.
Simplified radicals and fractions, not just decimals.
Whole numbers, decimals and fractions like 3/4.
Every quadratic equation ax² + bx + c = 0 with a ≠ 0 can be solved with x = (−b ± √(b² − 4ac)) / (2a). The expression under the square root, Δ = b² − 4ac, is called the discriminant and tells you what kind of roots to expect before you finish the calculation.
When all coefficients are fractions or whole numbers, the solver first scales the equation to whole numbers. That keeps the exact answer simple: perfect-square discriminants give rational roots, other positive discriminants give simplified radicals such as 3√2, and negative discriminants give complex roots with i.
Two different real roots: the parabola crosses the x-axis twice.
One repeated root: the vertex touches the x-axis.
Two complex roots: the parabola never meets the x-axis.
Yes. Type values such as 3/4 or -5/2, or decimals such as 0.25. The exact answer is still given as simplified fractions and radicals.
Then the equation is linear, bx + c = 0, and the solver gives its single solution x = −c/b.
The largest perfect square factor is taken out. For example √72 = √(36 × 2) = 6√2.
For accuracy, the decimal roots use a numerically stable form of the formula that avoids subtracting two nearly equal numbers.
For ax² + bx + c = 0 the roots add up to −b/a and multiply to c/a, which is a quick way to check your answer.
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