Quadratic Equation Calculator

What the Discriminant Decides Before You Solve Anything

The whole character of a quadratic is settled by one number: b² − 4ac, the part of the formula sitting under the square root. Work it out first and you know what kind of answer to expect, which saves solving an equation that was never going to give you what you wanted.

The second row is the useful one. Plenty of time is lost hunting for two whole numbers that multiply to give ac and add to give b, when the discriminant already proves no such pair exists. Thirty seconds spent on b² − 4ac tells you whether factorising is worth attempting.

Where the Formula Comes From

The quadratic formula is not an arbitrary thing to memorise. It is what you get when you complete the square on the general equation once, so that nobody has to do it again. Following the derivation makes the formula much harder to misremember.

Start from ax² + bx + c = 0 and divide through by a, which is legal precisely because a is not zero:

Two things fall out of that fourth line. The discriminant appears naturally as the thing being square-rooted, which is why its sign governs everything. And −b/2a turns up as the centre of the expression, which is exactly why the vertex of every parabola sits at x = −b/2a and why the two roots are always symmetric about it.

Checking an Answer Without Solving It Twice

There is a check that costs two pieces of mental arithmetic and catches almost every slip. For any quadratic, the two roots must add up to −b/a and multiply to give c/a. Those relationships hold whatever the roots are, and they do not depend on how you found them.

Take 2x² − 3x − 5 = 0, which this calculator opens with. The roots come out as 2.5 and −1. Their sum is 1.5, and −b/a is −(−3)/2 = 1.5. Their product is −2.5, and c/a is −5/2 = −2.5. Both match, so the answer is almost certainly right.

The check is strongest where mistakes are most likely: a dropped minus sign will nearly always break the sum, and a misread coefficient will break the product. It is worth doing even when you trust the arithmetic, because it takes longer to read this sentence than to perform.

The Four Mistakes That Produce Confident Wrong Answers

A quadratic gone wrong rarely announces itself. The answer looks like an answer — two plausible numbers — and nothing flags the error. Four causes account for most of them.

Not rearranging to zero. x² + 3x = 10 has c = −10, not 10, because the 10 has to cross the equals sign first. Solving with c = 10 gives complex roots for an equation whose real answers are 2 and −5.

Losing the sign of b. The formula opens with −b. When b is already negative, that becomes a plus, and entering 7 where the equation says −7 changes both roots.

Dividing before finishing the numerator. The 2a divides the whole of −b ± √(b² − 4ac), not just the square root. Written on one line without brackets this is easy to do and gives an answer that fails the sum check immediately.

Assuming a is 1. When the x² term has a coefficient it appears in three places — inside the discriminant, in the denominator, and in the factorised form. Halving your attention to it is enough to lose it in one of them.

If the arithmetic itself is the part that goes wrong rather than the method, the formula library has the quadratic formula and its discriminant written out with the symbols named, and the percentage calculator handles the proportion work that word problems usually wrap around a quadratic.

Quadratics Outside the Textbook

Quadratics describe anything where a quantity depends on the square of something else, which happens more often than the word "quadratic" suggests.

Anything thrown or dropped follows one, because gravity acts on time squared: height is a parabola against time, the vertex is the highest point, and the positive root is when it lands. Area problems produce them whenever a length appears twice — a path of uniform width around a lawn, a margin around a page, a fold in a sheet of metal. In business, revenue against price is quadratic whenever raising the price reduces the number sold, which is why the vertex, not a root, is the answer to "what should I charge".

That last case is worth noticing, because it is the one where people solve the wrong thing. The roots tell you where profit is zero. The vertex tells you where it is greatest. Both come from the same three coefficients, and this calculator reports both.

A quadratic is usually one step inside a longer question rather than the whole of it. Matrices, averages, fractions and the rest of the school syllabus are handled by the other math calculators, and each one shows its working the same way this page does.

Quadratic Equation Questions