Solving a quadratic equation
A quadratic equation has a squared term as its highest power. Its graph is a parabola, and the solutions (roots) are the x-values where the parabola meets the x-axis. Projectile height, area problems, profit curves and stopping distances all lead to quadratics. Factoring works only when the roots are tidy; the quadratic formula always works.
The formulas
x = (−b ± √(b² − 4ac)) / (2a) Discriminant: D = b² − 4ac Vertex: h = −b / (2a), k = c − b² / (4a) Vieta: x₁ + x₂ = −b / a, x₁ · x₂ = c / a
Worked example
Solve 2x² + 3x − 2 = 0: a = 2, b = 3, c = −2. D = 3² − 4·2·(−2) = 9 + 16 = 25, √D = 5.
x = (−3 ± 5) / 4, so x₁ = −8/4 = −2 and x₂ = 2/4 = 0.5. Check with Vieta: −2 + 0.5 = −1.5 = −b/a, and −2 × 0.5 = −1 = c/a. Factored: 2(x + 2)(x − 0.5).
For x² + 2x + 5 = 0, D = 4 − 20 = −16, so x = (−2 ± 4i) / 2 = −1 ± 2i.
Notes
- If a = 0 the equation is linear (bx + c = 0), not quadratic; the calculator tells you x = −c/b instead.
- Results are shown to 10 significant digits; integer and simple fractional roots are exact.
Sources: OpenStax, College Algebra 2e, §2.5 Quadratic Equations; W. H. Press et al., Numerical Recipes, 3rd ed., §5.6 (stable root formula).
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / 2a. The ± gives the two solutions. It works for every quadratic, including ones that do not factor nicely.
What does the discriminant tell me?
The discriminant D = b² − 4ac decides the kind of roots before you finish solving. D > 0: two different real roots (the parabola crosses the x-axis twice). D = 0: one repeated real root (it touches the axis at the vertex). D < 0: two complex conjugate roots (it never reaches the axis).
How do I enter an equation like 3x² = 5x − 2?
Move everything to one side first: 3x² − 5x + 2 = 0, so a = 3, b = −5, c = 2. A missing term has a coefficient of 0 — x² − 9 = 0 is a = 1, b = 0, c = −9.
Why is the answer slightly different from my hand calculation?
When b² is much larger than 4ac, the textbook formula subtracts two nearly equal numbers and loses digits. This calculator uses the algebraically identical form x₁ = q/a, x₂ = c/q with q = −½(b + sign(b)√D), which keeps full precision for the small root.