Quadratic Equation Solver

How the solver works

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its solutions, called roots, are the x-values where the parabola y = ax² + bx + c crosses the x-axis. The solver reads your a, b and c and returns:

• the discriminant D = b² − 4ac and what its sign means;
• the roots: as exact fractions when they are rational, in simplified square-root form and as decimals when they are not, or as a complex pair p ± qi when D is negative;
• the vertex (turning point), the axis of symmetry and the y-intercept;
• the factored form whenever the roots are rational, such as (2x + 1)(x − 2);
• each step of the quadratic formula with your numbers filled in, and a small sketch of the parabola.

Decimal coefficients are taken exactly as typed, so 0.1 means one tenth, not the nearest binary fraction. If a is 0 the equation has no x² term; the solver says so and solves the linear equation bx + c = 0 instead.

The quadratic formula and the discriminant

a, b, c

the coefficients of x², x and the constant term

D = b² − 4ac

the discriminant

±

one root uses +, the other −

The discriminant decides what kind of roots there are:

• D > 0: two different real roots. If D is a perfect square (of a rational number) they are rational and the quadratic factors neatly.
• D = 0: one repeated root, x = −b ÷ 2a. The parabola just touches the x-axis at its vertex.
• D < 0: no real roots. The square root of a negative number is imaginary (√−1 = i), so the roots are the complex pair −b/2a ± (√|D|/2a)i. The parabola never meets the x-axis.

The vertex sits halfway between the roots, at h = −b ÷ 2a, and its height is k = c − b² ÷ 4a. The vertical line x = h is the axis of symmetry, and the curve crosses the y-axis at (0, c).

Accuracy. When b² is much larger than 4ac, √D is almost equal to |b|, and the root that computes −b + √D (or −b − √D when b is negative) subtracts two nearly equal numbers, losing most of its digits. The solver computes the root of larger size first as q ÷ a, with q = −½(b + sign(b)·√D), and the other as c ÷ q, which is algebraically the same but has no such subtraction. For x² + 100,000,000x + 1 = 0 the textbook formula in ordinary double-precision arithmetic gives the small root about 25% wrong; the rearranged one gives −0.00000001 to full precision.

Worked example: 2x² − 3x − 2 = 0

Here a = 2, b = −3 and c = −2.

1. Discriminant: D = (−3)² − 4 × 2 × (−2) = 9 + 16 = 25. It is positive, so there are two real roots.
2. √25 = 5, a whole number, so the roots are rational.
3. x = (3 ± 5) ÷ 4, so x₁ = (3 − 5) ÷ 4 = −1/2 and x₂ = (3 + 5) ÷ 4 = 2.
4. Factored form: each root p/q gives a factor (qx − p), so 2x² − 3x − 2 = (2x + 1)(x − 2). Multiply it out to check: 2x² − 4x + x − 2.
5. Vertex: h = 3 ÷ 4 = 0.75 and k = −2 − 9 ÷ 8 = −3.125, so the lowest point of the parabola is (0.75, −3.125). The y-intercept is (0, −2).

Irrational roots. For x² − 2x − 1 = 0, D = 4 + 4 = 8 = 2² × 2, so x = (2 ± 2√2) ÷ 2 = 1 ± √2, about −0.4142135624 and 2.4142135624.

Complex roots. For x² + 2x + 5 = 0, D = 4 − 20 = −16 and √−16 = 4i, so x = (−2 ± 4i) ÷ 2 = −1 ± 2i.

Tips and common mistakes

• Move everything to one side first. For 3x² = 5x − 2, rewrite it as 3x² − 5x + 2 = 0 before reading off a = 3, b = −5, c = 2.
• Keep the signs. The most common slip is squaring a negative b without brackets: (−3)² is 9, not −9.
• Divide by 2a, not just 2. The whole numerator −b ± √D is divided by 2a.
• Check with the roots. The two roots always add up to −b ÷ a and multiply to c ÷ a. For the example: −1/2 + 2 = 3/2 and −1/2 × 2 = −1 = −2 ÷ 2.
• Fractions in the answer can be worked with in the fraction calculator, and powers, roots and other functions in the scientific calculator.

Limits

Coefficients can be any numbers from −10¹⁵ to 10¹⁵, including decimals. Irrational roots are shown to about 10 digits, and their simplified square-root form is given when the discriminant, scaled to whole numbers, is below about 9 × 10¹⁵; above that only the decimals are shown. The sketch of the parabola is a rough picture for orientation, not a scaled graph.