Quadratic equation solver
Enter a, b and c. The page solves ax² + bx + c = 0 and writes the discriminant and each root.
- Real or complex roots
- Free, no sign-up
a x² + b x + c = 0
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How to use it
- Enter a, b and c for ax² + bx + c = 0.
- Read the discriminant and whether the roots are real or complex.
- Follow the arithmetic down to the simplified roots.
What the discriminant decides
The discriminant is D = b² − 4ac. A positive D gives two different real roots, (−b + √D) / (2a) and (−b − √D) / (2a). Zero gives one repeated real root, −b / (2a). A negative D gives a complex pair with real part −b / (2a) and imaginary part √|D| / (2a). When a, b and c are whole numbers and D is a perfect square, the working stays in whole numbers and fractions.
If a is 0 and b is not, the equation is linear and the page solves x = −c / b. Square roots of whole numbers are simplified, so √8 appears as 2√2. The sample equation x² − 5x + 6 = 0 has discriminant 1 and roots 2 and 3.
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Questions
Does it show complex roots?
Yes. When the discriminant is negative the roots are written as a real part plus or minus an imaginary part, such as −1 + 2i.
What is the worked example?
a = 1, b = −5 and c = 6 is x² − 5x + 6 = 0. D = 25 − 24 = 1, and the roots are 3 and 2.
Can a be zero?
Yes. The equation is then linear, and the working solves it in one division. If a and b are both zero, the page says whether every number works or none does.
Will fractions stay exact?
Whole-number a, b and c keep an exact fraction when the square root is a whole number. Other inputs are shown as decimals.