Quadratic Formula Calculator

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are known numbers (the coefficients) and a ≠ 0. The condition a ≠ 0 is essential: if a were zero the x² term would vanish and you would be left with the linear equation bx + c = 0, which is not quadratic. Graphically, the left-hand side traces a parabola, and solving the equation means finding where that parabola crosses the horizontal x-axis. Those crossing points are called the roots (or solutions, or zeros) of the equation. This calculator takes your three coefficients, applies the quadratic formula, and returns the exact roots together with a full step-by-step breakdown.

The quadratic formula is the single most reliable method for solving these equations because it works for every quadratic, whether or not it factors neatly. Students preparing for CBSE, ICSE and state-board exams, JEE and other entrance tests, as well as engineers, physicists and anyone working with projectile motion, areas, or optimisation problems will reach for it constantly.

The Quadratic Formula

The formula is derived by completing the square on ax² + bx + c = 0. The plus-or-minus (±) symbol is what gives a quadratic its two roots: one uses the plus sign and the other the minus sign. The quantity under the square root, b²−4ac, is so important that it has its own name — the discriminant.

The Discriminant (b²−4ac)

The discriminant, often written as D or Δ, tells you the nature of the roots before you finish solving:

Positive: Two distinct real roots (parabola crosses x-axis twice)

Zero: One repeated real root (parabola touches x-axis once)

Negative: Two complex roots (parabola doesn't cross x-axis)

Worked Example

Solve x² − 5x + 6 = 0. Here a = 1, b = −5 and c = 6. First compute the discriminant: D = b²−4ac = (−5)² − 4(1)(6) = 25 − 24 = 1. Because D is positive, expect two distinct real roots. Substitute into the formula: x = (−(−5) ± √1) / (2·1) = (5 ± 1) / 2. The plus branch gives x = (5 + 1)/2 = 3, and the minus branch gives x = (5 − 1)/2 = 2. So the two roots are x = 3 and x = 2. You can verify by factoring: x² − 5x + 6 = (x − 2)(x − 3), which confirms the answer.

Tips for Using the Formula

Always rearrange your equation into standard form first, moving every term to one side so the other side equals zero — only then can you read off a, b and c correctly. Watch the sign of b carefully; in the example above b was −5, not 5, and −b became +5. A quick sanity check is Vieta's formulas: the two roots should add up to −b/a and multiply to c/a. For x² − 5x + 6 the roots 2 and 3 sum to 5 (= −(−5)/1) and multiply to 6 (= 6/1), so the answer is consistent.

Common Mistakes

The most frequent error is dividing only part of the numerator by 2a — remember the entire expression −b ± √D sits over 2a. Another is forgetting that 2a (not a) goes in the denominator. Many learners also drop the ± and report just one root, or mishandle a negative discriminant by trying to take the square root of a negative number instead of recognising it as a complex (imaginary) result. Finally, double-check that a is genuinely non-zero; if it is zero, solve the equation as a linear one instead.

Other Ways to Solve a Quadratic

The quadratic formula is universal, but it is not always the quickest route. Factoring is fastest when the roots are simple integers or fractions: write the expression as a product such as (x − 2)(x − 3) = 0 and set each bracket to zero. Completing the square rewrites the equation in the form (x − h)² = k and is the very method the formula is derived from; it is also how you find the vertex of the parabola. For some equations graphing or simple square-rooting (when b = 0, as in x² − 4 = 0) is enough. The advantage of the formula is that it never fails — it handles awkward, irrational and complex roots that factoring cannot reach — which is why it is the safe default whenever a method is not obvious.

Where Quadratics Are Used

Quadratic equations are far from an abstract exercise. In physics, the height of a thrown or launched object under gravity follows a quadratic in time, so solving one tells you when a ball lands or reaches a target height. In business, profit and revenue models are frequently quadratic, and the vertex of the parabola pinpoints the price or quantity that maximises profit. Engineers use them for areas, trajectories and parabolic shapes such as satellite dishes and bridge arches, while the discriminant flags whether a real-world scenario has a feasible solution at all. A negative discriminant in such a problem usually means the situation cannot physically happen — for instance, the object never actually reaches the height in question.