Quadratic Formula (Positive Root)

The quadratic formula solves every equation of the form ax² + bx + c = 0 in one stroke, and this page returns the plus branch — the root you get by adding the square root of the discriminant. Its twin, the minus branch, is a separate calculator here because a solver returns one number at a time; run both with the same a, b and c to see the complete pair. Worked example: 2x² + 3x − 5 = 0 gives b² − 4ac = 9 + 40 = 49, so x = (−3 + 7)/4 = 1 here and (−3 − 7)/4 = −2.5 on the minus branch, and indeed 2(1)² + 3(1) − 5 = 0.

The method is older than the notation. A Babylonian scribe working the clay tablet BM 13901 around 1800 BCE handled "I added the area and the side of my square and got 0;45" by halving the coefficient, squaring it, and adjusting — completing the square, three and a half millennia before anyone wrote a formula down. Around 820 CE in Baghdad, al-Khwārizmī's al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala gave the word al-jabr ("restoration") to the whole subject, though with no negative numbers available he had to treat six separate cases. The trap to avoid: a must not be zero (the equation is then linear), and if the discriminant comes out negative there is no real root at all — the parabola never touches the x-axis. Watch the signs, too: with b negative, −b is positive.