Quadratic Solver

How to use Quadratic Solver

1. Enter the coefficients a, b and c — a cannot be zero, or the equation is linear.
2. Read the discriminant to see what kind of roots to expect.
3. Read the roots and the vertex, with the working shown.
4. If a is 0 the equation is linear, not quadratic, and has a single root at −c/b.
5. Complex roots are returned as a conjugate pair when the discriminant is negative.
6. The sum of the roots is −b/a and their product is c/a — a quick way to check an answer.

The discriminant tells you the shape of the answer

b² − 4ac decides everything before you compute a root. Positive means two distinct real roots, so the parabola crosses the x-axis twice. Zero means one repeated root, where the curve touches the axis at its vertex. Negative means no real roots — the parabola misses the axis entirely and the solutions are a complex conjugate pair. Checking the discriminant first tells you what kind of answer to expect and catches an arithmetic slip before you finish.

Completing the square, and where the formula comes from

The quadratic formula is not arbitrary: it is what you get by completing the square on the general form. Divide through by a, move c across, add (b/2a)² to both sides to make a perfect square, and take the root. That derivation is worth doing once, because it explains the ±, explains why the vertex sits at x = −b/2a, and makes the formula something you can reconstruct rather than something you have to remember.

The vertex is usually the interesting answer

Roots tell you where a quantity is zero. In most word problems — projectile height, profit against price, area against a fixed perimeter — the question is really about the maximum or minimum, which is the vertex at x = −b/2a. If a is positive the parabola opens upward and the vertex is a minimum; if negative it opens downward and the vertex is a maximum. Many students find the roots correctly and then answer the wrong question.

Where the formula loses precision

When b² is much larger than 4ac, one root involves subtracting two nearly equal numbers, and floating-point arithmetic loses significant digits doing it — catastrophic cancellation. Numerical libraries avoid it by computing the well-conditioned root with the formula and the other as c/(a·x₁), using the relationship that the product of the roots is c/a. It rarely matters for homework and matters a great deal in simulation code.