Math Formula Reference and Solver (quadratic)
100% PrivateEducation & Research ToolsSolve quadratic equations step by step, with a formula reference beside the solver.
Runs entirely in your browser. Your input is not uploaded to Ovatly or anywhere else.
Solves ax^2 + bx + c = 0 and shows the working, not just the answer.
Formula reference
Solves ax^2 + bx + c = 0. The part under the root, b^2 - 4ac, is the discriminant: positive means two real roots, zero means one repeated root, negative means no real roots.
For right-angled triangles only. c is the hypotenuse, the side opposite the right angle, and is always the longest side.
Pythagoras applied to a coordinate grid: the horizontal and vertical gaps between the points form the two shorter sides.
Rise over run. A vertical line has no defined slope because the run is zero.
r is the radius, half the diameter. pi is about 3.14159.
The height must be measured at a right angle to the base, not along a slanted side.
Uses the radius. If you are given the diameter, halve it first.
The volume is the area of the circular base times the height. The surface area includes both round ends.
P is the starting amount, r the yearly rate as a decimal, n the number of compounding periods per year, t the years. Listed as a reference formula; the solvers on this page do not compute it.
A negative answer means the value went down. Divide by the original value, not the new one.
- 1Pick a solver: quadratic, Pythagoras, two points, or area and volume.
- 2Type the values it asks for (for Pythagoras, leave the side you want blank).
- 3Read the answer together with the numbered working steps.
- 4Scroll down for the formula reference cards that explain each formula in words.
With a = 1, b = -5 and c = 6, the discriminant is D = 25 - 24 = 1. Because D is positive there are two real roots: x = 3 and x = 2. The vertex of the parabola is at (2.5, -0.25), and the graph sketch shows the curve crossing the x-axis at 2 and 3.
- When the discriminant is negative the solver says plainly that there are no real roots and gives the complex roots instead; school courses that have not covered complex numbers can stop at "no real roots".
- The graph is a sketch in a window centered on the vertex, meant for checking shape and crossings, not for measuring values off the axes.
- Pythagoras only applies to right-angled triangles, and the hypotenuse must be the longest side; the solver refuses impossible combinations instead of printing a wrong side.
- Results are shown to at most 10 significant digits, which is more precision than school work needs; the underlying calculation uses full double precision.
The vertex is the turning point of the parabola, and it sits exactly halfway between the roots. Seeing it helps you sketch the curve and check whether your roots are plausible.
About this tool
The Math Formula Reference and Solver combines worked solvers with a formula reference. The quadratic solver shows the discriminant, the roots and the vertex with every step, plus a graph sketch; there are also solvers for right-angled triangles, slope and distance between two points, and areas and volumes, each showing its working. It is aimed at students who need to see the method, not just the answer.
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