Quadratic formula calculator

Type the equation the way it's written, such as 2x^2 + 3x - 2 = 0. Write x² as x^2, or paste the ² sign. Terms can be on both sides of the = sign, and you can also type just the three numbers a, b and c, separated by commas.

Whole-number equations get exact roots, in fractions or square roots, with the decimals beside them. When the roots are fractions, the calculator also shows the factored form.

For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / 2a. The ± gives the two roots: one with the plus and one with the minus.

The part under the square root, b² − 4ac, is the discriminant. If it's positive there are two real roots and the parabola crosses the x-axis twice. If it's 0 there's one repeated root and the parabola just touches the axis. If it's negative there are no real roots, only two complex ones.

The curve is y = ax² + bx + c. The roots are where it crosses the x-axis, and the vertex, its lowest or highest point, sits halfway between them. The dashed line through the vertex is the axis of symmetry.

A positive a opens the parabola upwards. A negative a flips it.

The bigger a is, the narrower the curve, because the x² term grows faster and pulls both arms up steeply from the vertex, while a small a such as 0.1 gives a wide, shallow bowl.

Babylonian clay tablets already set problems that come down to quadratic equations. Around 628 AD the Indian mathematician Brahmagupta described a rule for one root in words, and in the 9th century al-Khwarizmi solved them algebraically.

Simon Stevin first obtained a formula covering every case, in 1594. That's almost a thousand years after Brahmagupta.