How to use it
First put your equation in standard form, \(ax^2 + bx + c = 0\), with everything on the left and 0 on the right. Then enter a, b and c, signs included. A missing term has a coefficient of 0, so for \(x^2 - 9 = 0\) you enter \(a = 1\), \(b = 0\), \(c = -9\). The equation you typed is shown above the boxes, which makes a wrong sign easy to spot.
What the calculator does
It starts with the discriminant, \(b^2 - 4ac\), written \(\Delta\) here. Its sign tells you what kind of solutions to expect:
- positive: two different real solutions, from the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\);
- zero: one repeated solution, \(x = -\frac{b}{2a}\);
- negative: no real solutions. The calculator still shows the two complex solutions, written with \(i\).
Answers are exact. When the discriminant isn't a perfect square, the radical is put in simplest radical form, like \(\sqrt{48} = 4\sqrt{3}\), and a decimal approximation is listed next to it.
Shortcuts and checks
Not every quadratic needs the formula. With no constant term, factor out x: \(3x^2 - 6x = 0\) becomes \(3x(x - 2) = 0\), so \(x = 0\) or \(x = 2\). With no x term, isolate \(x^2\) and take square roots, remembering the \(\pm\). The calculator points out these shortcuts when they apply.
At the end you'll find the factored form \(a(x - x_1)(x - x_2)\), a check using the sum and product of the solutions (\(-\frac{b}{a}\) and \(\frac{c}{a}\)), and the vertex of the parabola \(y = ax^2 + bx + c\). Graph it with the function grapher and you'll see the solutions are exactly the x-intercepts.