Quadratic Formula Calculator

Every quadratic equation — any equation that can be arranged into ax² + bx + c = 0 — can be solved by one universal formula, whether or not the expression factors neatly. That reliability is the point. Factoring is quicker when it works, but plenty of quadratics refuse to factor over the rationals, and their roots are irrational or even complex. The quadratic formula calculator applies the formula directly: enter the three coefficients and it returns the roots, handling the discriminant, the ± split and the square root without the sign errors that plague hand working.

Quick Answer

The roots of ax² + bx + c = 0 are x = [−b ± √(b² − 4ac)] / (2a). The quantity under the root, D = b² − 4ac — the discriminant — decides what kind of roots you get: positive means two real roots, zero means one repeated root, negative means two complex roots.

The Shape of a Quadratic Equation

A quadratic is a second-degree polynomial: one squared term, one plain term, one constant. Graph it and you get a parabola, opening upward when a is positive and downward when a is negative. Solving the equation means finding where that parabola crosses the x-axis, which is why the number of real roots is at most two — a parabola can cross a line at most twice.

The three coefficients each steer the curve. The value a controls width and direction; b shifts the axis of symmetry away from the y-axis; c is the height where the parabola crosses the vertical axis. The formula itself comes from completing the square on the general equation — it is not a trick to memorise but the completed-square solution written once, in full generality, so nobody has to redo it. The discriminant deserves its starring role: it is the part under the square root, so its sign alone determines whether the square root is real, zero, or imaginary, and with it the entire character of the answer.

How to Use the Quadratic Formula Calculator

1. Rearrange your equation so one side is zero and the terms sit in descending order: x² term, x term, constant.
2. Read off a, b and c, keeping the sign attached to each coefficient — if a term is subtracted, its coefficient is negative.
3. If a term is missing, its coefficient is 0; enter that rather than skipping it.
4. Enter the three coefficients and run the calculation.
5. Read both roots; when the discriminant is negative, expect the answer in complex form, a real part plus an imaginary part.
6. Verify at least one root by substituting it back into the original equation — the result should be zero (or within rounding of it).

Worked Example: Solving 2x² − 4x − 6 = 0

Here a = 2, b = −4 and c = −6. Start with the discriminant: D = b² − 4ac = (−4)² − 4(2)(−6) = 16 + 48 = 64. Positive, so expect two real roots, and √64 = 8 exactly.

Now the formula: x = [−(−4) ± 8] / (2 × 2) = (4 ± 8)/4. Taking the plus sign: x = 12/4 = 3. Taking the minus: x = −4/4 = −1. Check both in the original equation: at x = 3, 2(9) − 12 − 6 = 0 ✓; at x = −1, 2(1) + 4 − 6 = 0 ✓. Both roots verified.

• x² − 6x + 9 = 0 → D = 0, one repeated root: x = 3
• x² + 2x + 5 = 0 → D = −16, complex roots: x = −1 ± 2i
• x² − 5x + 6 = 0 → D = 1, roots x = 2 and x = 3
• 3x² + x − 2 = 0 → D = 25, roots x = 2/3 and x = −1

Checking Roots With the Sum and Product

Two relationships, known as Viète's formulas, let you audit any pair of roots in seconds: the roots of ax² + bx + c = 0 always add to −b/a and multiply to c/a. In the worked example, the roots 3 and −1 sum to 2, and −b/a = 4/2 = 2; their product is −3, and c/a = −6/2 = −3. Both agree, so the pair must be right. These checks also locate the roots on the graph without plotting: the parabola's axis of symmetry sits at x = −b/(2a), exactly halfway between the roots. For the example that axis is x = 1, the vertex value there is 2 − 4 − 6 = −8, and sure enough the roots 3 and −1 stand two units either side of the axis — a symmetry you can use to catch an asymmetric arithmetic slip on sight.

Reading the Discriminant Before You Solve

Checking D first prevents the commonest errors, because it tells you what shape of answer to expect. The most frequent mistakes happen inside it: when b is negative, b² is still positive; when c is negative, the term −4ac becomes a plus, and many students subtract anyway — in the worked example that single slip changes D from 64 to −32 and the whole answer collapses. Forgetting the ± is next: dividing only by the plus branch throws away a root. And if a turns out to be 0, you never had a quadratic at all — the equation is linear and needs no formula. When D is a perfect square, the roots are rational and factoring would have worked; when it is not, the surd form the formula produces is the exact answer, and decimal approximations should be labelled as such.

Other Ways to Attack a Quadratic

The formula never fails, but it is not always the fastest tool. When the numbers are friendly, the Factor Calculator breaks the expression into brackets and the roots fall out by inspection. The Remainder Calculator connects through the factor theorem — a value is a root exactly when it leaves remainder zero. All three live in the math section for the many occasions when an equation needs a second opinion.