Quadratic Equation Calculator

How to use the calculator

1. Rewrite your equation in standard form, with everything on one side and 0 on the other. For example, x² = 3x + 10 becomes x² − 3x − 10 = 0.
2. Enter a (the x² coefficient), b (the x coefficient) and c (the constant). Use negative signs where needed and 0 for a missing term. Decimals are fine.
3. Read the solutions at the top. The individual roots x₁ and x₂ and the factored form appear only when the roots are real.

The coefficient a can't be 0: without an x² term the equation is linear, not quadratic.

How the calculator solves it

Everything starts with the discriminant:

Δ = b² − 4ac

• Δ > 0: two distinct real roots, x = (−b ± √Δ) / 2a.
• Δ = 0: one repeated real root, x = −b / 2a; the parabola touches the x-axis at its vertex.
• Δ < 0: two complex conjugate roots, x = −b/2a ± (√−Δ / 2|a|) i; the parabola never crosses the x-axis.

When b is large compared with a and c, the textbook formula subtracts two nearly equal numbers and loses accuracy. The calculator avoids this by first computing q = −(b + sign(b)·√Δ) / 2, then taking x₁ = q/a and x₂ = c/q. Both roots stay accurate.

The other outputs follow directly from the coefficients:

• Vertex: (−b/2a; −Δ/4a), the turning point of y = ax² + bx + c.
• Vieta's formulas: sum of roots = −b/a, product = c/a.
• Factored form: a(x − x₁)(x − x₂), or a(x − x₁)² for a double root.

Displayed values are rounded to 10 significant digits. The text results (solutions, vertex, factored form) use a true minus sign and write every number out in plain decimal form, with no exponent notation and no thousands separators. When one root is 0, the factored form shows a bare x factor, such as −16x(x − 4).

Worked examples

1. Two real roots: x² − 3x − 10 = 0

With a = 1, b = −3, c = −10, the discriminant is 49. The roots are x₁ = −2 and x₂ = 5, so the factored form is (x + 2)(x − 5). The vertex is (1.5; −12.25), the sum of the roots is 3 and the product is −10.

2. A ball thrown upward

A ball leaves a 48-foot rooftop at 32 ft/s. Its height is h(t) = −16t² + 32t + 48. Solve −16t² + 32t + 48 = 0: the roots are t = −1 and t = 3. Time can't be negative, so the ball lands after 3 seconds. The vertex (1; 64) says it peaks at 64 feet after 1 second.

3. Complex roots: x² + 2x + 5 = 0

Here Δ = −16, so there are no real solutions. The calculator returns x = −1 ± 2i. The vertex (−1; 4) sits above the x-axis on an upward-opening parabola, which is why the graph never crosses it.

4. Where accuracy matters

For x² + 100,000,000x + 1 = 0, the small root is about −0.00000001. Evaluated directly in standard double-precision arithmetic, the textbook formula returns about −0.00000000745, roughly 25% off. The stable method used here gets both roots right and displays them as x₁ = −100000000, x₂ = −0.00000001.

Checking your answer

Use Vieta's formulas as a quick check: add your two roots and compare with −b/a, multiply them and compare with c/a. For word problems, decide which root makes sense. Negative times or lengths are usually discarded, as in the ball example. Irrational roots such as 1.618033989 are rounded; if your teacher wants an exact answer, keep the radical form (1 + √5)/2.