One formula solves every quadratic, and its inside-out twin is called citardauq
Any equation with a squared unknown, a plain unknown and a constant can be cracked with a single expression: negative b, plus or minus the square root of b squared minus four a c, all over two a. A lesser-known cousin puts that square root in the denominator and is jokingly named citardauq, quadratic spelled backwards.
A quadratic equation sets a times x squared, plus b times x, plus c equal to zero, where a is not zero and the coefficients may be real or complex. The quadratic formula gives its solutions, known as roots or zeros, in closed form. The plus-or-minus sign produces two answers, and other methods, such as completing the square, reach exactly the same pair.
Everything hinges on the quantity under the square root, b squared minus four a c, called the discriminant. With real coefficients, a positive discriminant means two distinct real roots, zero means a single repeated root, and a negative value means no real roots but two complex ones that are conjugates of each other. Geometrically, real roots mark where the parabola traced by the quadratic crosses the horizontal axis, and the formula also locates the curve's axis of symmetry.
The standard derivation begins by dividing through by a, then moving the constant to the other side. Adding the square of half the new middle coefficient turns the left side into a perfect square. Taking square roots of both sides and isolating x delivers the familiar result.
Variants trade one arrangement for another. Dividing the original equation by two a first and substituting yields a version that reuses an intermediate quantity, trimming a little arithmetic. The version first mentioned by Giulio Fagnano writes each root as two c divided by negative b minus or plus the square root of the discriminant, so the radical sits below the line. It lists the same two roots in the same order, simply rearranged.
Source: Quadratic formula
