Solving equations with an x squared term
A quadratic equation has the standard form ax2 + bx + c = 0, where a is not zero. Its graph is a parabola — a U-shape (or an upside-down U when a is negative) — and solving the equation means finding where that parabola crosses the x-axis. Those crossing points are the roots. Depending on the numbers, a parabola can cross twice, touch once, or miss the axis entirely, and this calculator handles all three cases, including the complex-number roots that appear when it misses.
The quadratic formula and the discriminant
The quantity inside the square root, D, is called the discriminant, and it decides the kind of answer before you finish the calculation. If D is positive, its square root is a real number and the ± gives two different real roots. If D is zero, the ± adds and subtracts nothing, leaving one repeated root. If D is negative, the square root involves the imaginary unit i and the roots are a complex conjugate pair — the real parts match and the imaginary parts are opposites. The step-by-step panel substitutes your actual coefficients into the formula so you can follow every stage.
The vertex and the axis of symmetry
The parabola is symmetric around a vertical line through its turning point, the vertex. The vertex sits at x = −b ÷ (2a), and putting that x back into the equation gives its y value. That same x is the axis of symmetry, and it is also exactly halfway between the two roots when they are real — a useful check. The y-intercept is simply c, the value when x = 0. The points table in the detail section lists y for x values around the vertex, so you can sketch the curve and see whether it crosses, touches or misses the axis.
Where quadratics show up
Projectile motion is the classic case: the height of a thrown ball follows a quadratic in time, and solving it finds when the ball lands. Quadratics also appear in area problems (a rectangle with a fixed perimeter), braking distances, revenue curves in business, and the design of satellite dishes and headlights, whose shapes are parabolas. If you can rearrange an equation into the standard form — everything on one side, zero on the other — its coefficients a, b and c are all this calculator needs.
Worked example
Solve x2 − 5x + 6 = 0. Here a = 1, b = −5, c = 6, so D = 25 − 24 = 1. The roots are (5 ± 1) ÷ 2, giving x = 3 and x = 2. The vertex is at x = 2.5, where y = −0.25, neatly halfway between the roots.
How to use this calculator
- Rearrange your equation so everything is on one side and zero is on the other.
- Read off a (the x squared coefficient), b (the x coefficient, keeping its sign) and c (the constant).
- Enter a, b and c — remember b and c can be negative.
- Read the first root in the green panel and the full breakdown below it.
- Use the discriminant to understand the answer: positive means two roots, zero means one repeated root, negative means complex roots.
Checking your roots by factoring or substitution
Two quick checks catch most sign errors. First, substitute each root back into the original equation — it should produce zero (allowing for tiny rounding). Second, for real roots, the sum of the roots should equal −b ÷ a and their product should equal c ÷ a: here the defaults give roots 2 and 1, whose sum is 3 = −(−3) ÷ 1 and whose product is 2 = 2 ÷ 1. When roots are whole numbers, the equation also factors neatly — x2 − 3x + 2 = (x − 2)(x − 1) — and spotting that pattern is often the fastest hand method. The Scientific Calculator can verify a substitution directly by typing the equation value at a root.