What it is
A quadratic equation has the form ax2 + bx + c = 0, where a, b and c are numbers and a is not zero. The calculator solves a quadratic equation online using the discriminant, shows the solution steps and plots the parabola y = ax2 + bx + c — the roots are the points where the parabola crosses the x axis.
How it is solved
Quadratic equations are solved using the discriminant (D):
D = b2 − 4ac
- If D > 0, the equation has 2 real roots. They are found by the formulas:
x1 = (−b − √D) / 2a,x2 = (−b + √D) / 2a. - If D = 0, the equation has 1 real root, found by the formula:
x = −b / 2a. - If D < 0, the equation has no real roots (the parabola does not cross the x axis; roots exist only among complex numbers).
Examples
| Equation | D | Roots |
|---|---|---|
| x2 − 5x + 6 = 0 | 25 − 24 = 1 | x1 = (5 − 1)/2 = 2, x2 = (5 + 1)/2 = 3 |
| 2x2 + 4x + 2 = 0 | 16 − 16 = 0 | x = −4/4 = −1 |
| x2 + 2x + 5 = 0 | 4 − 20 = −16 | no real roots |
Good to know
- Vieta's formulas: x1 + x2 = −b/a, x1·x2 = c/a. For x2 − 5x + 6: 2 + 3 = 5, 2·3 = 6 — a handy check.
- The vertex is at x = −b/2a; for a > 0 the parabola opens upward (minimum), for a < 0 downward (maximum).
- Incomplete equations are easier: for c = 0 factor out x (x(ax + b) = 0), for b = 0 use x2 = −c/a.
- If a = 0 the equation is linear, not quadratic: bx + c = 0, x = −c/b — the calculator will tell you so.
Where it is used
- Physics — uniformly accelerated motion: a body thrown up from 20 m at 15 m/s has h = 20 + 15t − 5t2; from 5t2 − 15t − 20 = 0 (D = 225 + 400 = 625) we get t = (15 + 25)/10 = 4 s until it lands (the other root, −1, does not fit).
- Geometry — sides of a rectangle from its area and perimeter, a radius from the area of a ring.
- Economics — break-even point and maximum profit when revenue or cost depends quadratically on volume.
- Engineering — ballistics, parabolic arches and spans, electric circuits (power delivered to a load).
- Studying — check homework and see the step-by-step solution with a graph.
How to use
- Enter the coefficients a, b and c into the equation fields (fractional and negative numbers work).
- Press "Execute" — the discriminant, the number of roots and the roots themselves appear with the values substituted into the formulas.
- Below, the parabola is plotted with its roots and vertex marked.