Slope between two points
Calculate slope, rise, run, intercept, line equation and distance from two points. Handle vertical lines and identical points explicitly.
Quadratic equation calculator with real and complex roots
Enter the coefficients of ax² + bx + c = 0. The calculator distinguishes two real roots, a repeated root and a conjugate complex pair. If a is zero, it solves the remaining linear equation instead of dividing by zero.
Free to use, with no sign-in. Inputs and results stay in this page and are not sent with the form. Download a result before leaving; this tool does not save your work.
Worked result
The roots are 2 and 3; the vertex is (2.5, -0.25).
- Calculate the discriminant, 1.
- A positive discriminant gives two distinct real roots.
- Substitute each root into the original expression to check it equals zero.
Solve x² − 5x + 6 = 0
Here a = 1, b = -5 and c = 6. Parentheses matter when squaring a negative coefficient.
Discriminant D = (-5)² − 4×1×6 = 1
x = (5 ± √1)/2
Roots: x = 2 and x = 3
Vertex x = 5/2 = 2.5
Vertex y = -0.25The roots are x-axis intersections of the parabola, while the vertex is its turning point. They describe different features of the same quadratic.
Use the discriminant before evaluating the roots
The sign of b²−4ac determines the root type for real coefficients. Zero gives one repeated real root; a negative value produces two complex conjugates. The axis of symmetry is x = -b/(2a).
For distinct real roots, the implementation uses a rearranged q form to reduce cancellation when subtracting nearly equal numbers, then obtains the other root from c/q. Results still use floating-point arithmetic and may be sensitive near a repeated root.
D = b² − 4ac
x = (-b ± √D)/(2a), a ≠ 0
vertex = (-b/(2a), -D/(4a))
When a = 0 and b ≠ 0: x = -c/bInput limits and common mistakes
Supply finite coefficients with absolute magnitude at most 1e100. This is a numeric solver; it does not accept an equation string, symbolic coefficients or arbitrary-precision arithmetic.
If both a and b are zero, c = 0 is an identity and nonzero c gives no solution. Near-zero coefficients and nearly repeated roots can be ill-conditioned. Check sensitive results with an appropriate high-precision or symbolic method.
Questions about quadratic equation solver
What if the discriminant is negative?
There are no real roots, but there are two complex conjugate roots. For x² + 1 = 0, they are i and -i.
What if a equals zero?
The equation is linear or constant. The calculator handles those cases separately.
Does a repeated root count as two roots?
It is one distinct root with multiplicity two. The discriminant is zero.
Sources and verification
These references explain the formulas or formats. The worked examples above were created for this tool and checked against independent calculations or format round trips. AI-assisted development does not imply a named human expert review.
- OpenStax: the quadratic formula — The formula and discriminant classification.
Published 2026-10-23. Examples are illustrations, not measurements of a service or claims about search rankings.
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Further reading
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