Quadratic Equation Solver

When Does the Ball Land? The Physics Problem That Made Quadratics Essential

A ball is thrown upward at 20 m/s from a height of 5 metres. Its height at time t follows h(t) = −5t² + 20t + 5. When does it hit the ground? Set h = 0: −5t² + 20t + 5 = 0. This is a quadratic equation — and the quadratic formula gives two solutions, one of which is the landing time. The negative root is mathematically valid but physically meaningless (it represents a time before the throw), so you take the positive one. This pattern — a physical situation producing a quadratic, with one root meaningful and one discarded — appears constantly in projectile motion, circuit analysis, structural engineering and economics.

This free quadratic equation solver handles all three root types — two real roots, one repeated root, and two complex roots — and shows the discriminant, vertex, axis of symmetry, parabola direction and factored form. Enter a, b, c and results appear instantly with a complete 6-step solution. No signup, no installation, nothing to install.

The Discriminant Is a Decision-Maker, Not Just a Number

Before solving, Δ = b² − 4ac tells you what kind of answer to expect. If Δ > 0, the parabola crosses the x-axis at two distinct points — two real solutions exist. If Δ = 0, the parabola just touches the x-axis — one repeated root, a unique solution. If Δ < 0, the parabola never reaches the x-axis — roots are complex numbers involving √(−1). In real-world problems, Δ < 0 often signals an impossible scenario: that ball thrown upward never reaches the target height you set.

For complex roots, the result is expressed as a ± bi where a is the real part and b the imaginary part — useful for electrical engineering problems involving impedance in AC circuits. The solver shows the discriminant prominently so you understand the nature of the solution before reading the roots themselves. This is the correct way to approach any quadratic: classify first, then compute.

Vieta's Shortcut — Verify Roots Without Substituting

Once you have roots x₁ and x₂, Vieta's formulas let you verify them instantly: x₁ + x₂ = −b/a and x₁ × x₂ = c/a. For x² − 5x + 6 = 0 with roots 2 and 3 — sum = 5 = −(−5)/1 ✓, product = 6 = 6/1 ✓. No substitution needed. This works even when roots are messy decimals and substituting back would be error-prone. On timed exams, this check takes under ten seconds and catches arithmetic mistakes before they cost marks.

The factored form a(x − x₁)(x − x₂) shown in the results is the form needed for quadratic inequalities. Once you know the roots and the sign of a, you can immediately determine where the parabola is positive or negative across all intervals. The vertex coordinates shown — (h, k) where h = −b/(2a) — give the minimum or maximum value of the quadratic function directly, which is the solution to profit maximisation and cost minimisation problems in economics and engineering optimisation.

Quadratics Appear Everywhere You Look

The shape of a satellite dish, suspension bridge cable, car headlight reflector, and telescope mirror are all parabolas — the geometric shape described by quadratic equations. Stopping distance in traffic safety studies is quadratic in speed (doubling speed quadruples stopping distance). Compound interest approximations, the trajectory of a football, the bending moment in a simply supported beam, and the area of a rectangle with fixed perimeter — all quadratic. For evaluating the parabola at specific points or working through trig-based variations, the Scientific Calculator handles those arithmetic steps. For equations with fractional coefficients, simplify first using the Fraction Calculator then enter the decimal equivalents here. All calculations run entirely in your browser — nothing is transmitted to any server.

✓Verified by ToollyX Team · Last updated June 2026