No real roots does not mean no roots: read the discriminant
The sign of b²−4ac tells you the type of roots before you finish the quadratic formula.
Quadratic equation →Three cases for real coefficients
For a nonzero a and real coefficients, a positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives a pair of nonreal complex conjugates. This classification concerns the equation, not whether a particular real-world problem accepts every mathematical solution. The calculator shows the discriminant so you can check which situation applies.
Compare three equations
For x²−4 = 0, the discriminant is 16 and roots are −2 and 2. For x²−4x+4 = 0, the discriminant is 0 and the repeated root is 2. For x²+4 = 0, the discriminant is −16 and the roots are 0±2i, where i² = −1. The last equation has no real root but still has complex roots.
Read the complex output correctly
When the tool returns a real part and an imaginary magnitude, combine them as real part plus or minus that magnitude times i. Do not treat the imaginary magnitude as another real root. Rounded coefficients near a zero discriminant can change the apparent root type; use the most accurate available input and avoid interpreting a tiny positive or negative computed value as proof of physical certainty. A repeated root is one numerical value with multiplicity two, not two different real solutions.
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