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Summary
A quadratic equation is a second-degree polynomial of the form ax^2 + bx + c = 0. These equations model parabolic trajectories, area optimization, projectile motion, and countless other phenomena. The quadratic formula provides exact solutions for any quadratic equation, and the discriminant reveals how many real solutions exist before you even solve it.
How it works
The calculator takes three coefficients (a, b, c) and computes:
- Roots (solutions) — the values of x where the parabola crosses the x-axis, found using the quadratic formula.
- Discriminant — the value under the square root that determines the nature of the roots: two real, one repeated, or two complex.
- Vertex — the highest or lowest point of the parabola, found at x = -b/(2a).
- Axis of symmetry — the vertical line x = -b/(2a) that divides the parabola into two mirror halves.
The discriminant
- D > 0 — two distinct real roots. The parabola crosses the x-axis at two points.
- D = 0 — one repeated real root (a “double root”). The parabola touches the x-axis at exactly one point.
- D < 0 — no real roots; two complex conjugate roots. The parabola does not cross the x-axis.
The formulas
Where
Where
Worked examples
Solve 2x^2 + 5x - 3 = 0
Identify coefficients
= a = 2, b = 5, c = -3
Calculate the discriminant
= 49
Apply the quadratic formula
= x = 2/4 or x = -12/4
Simplify
= Two real roots
Result
x = 0.5 and x = -3
Find the vertex of y = x^2 - 6x + 5
Identify coefficients
= a = 1, b = -6, c = 5
Calculate x-coordinate of vertex
= 3
Calculate y-coordinate of vertex
= -4
Result
Vertex is at (3, -4) -- a minimum since a > 0
Practical uses
- Projectile motion — the height of a thrown ball follows a parabolic path; the roots give when it hits the ground, and the vertex gives maximum height.
- Area optimization — maximizing the area of a rectangle with a fixed perimeter leads to a quadratic equation.
- Revenue modeling — if price and demand have a linear relationship, revenue (price times quantity) is quadratic, and the vertex gives maximum revenue.
- Engineering — parabolic shapes appear in bridges, satellite dishes, and headlight reflectors.
Assumptions & limitations
- a cannot be zero — if a = 0, the equation becomes linear (bx + c = 0), not quadratic. The calculator rejects a = 0.
- Complex roots — when the discriminant is negative, the roots involve imaginary numbers. The calculator displays these as a +/- bi where i = sqrt(-1).
- Floating-point precision — for very large or very small coefficients, rounding may introduce minor inaccuracies in the displayed roots. The calculator uses standard double-precision arithmetic.