Quadratic formula

In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way.

The general quadratic equation is

Here x represents an unknown, while a, b, and c are constants with a not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation, by inserting the former into the latter. With the above parameterization, the quadratic formula is:

Each of the solutions given by the quadratic formula is called a root of the quadratic equation. Geometrically, these roots represent the x values at which any parabola, explicitly given as y = ax² + bx + c, crosses the x-axis. As well as being a formula that will yield the zeros of any parabola, the quadratic equation will give the axis of symmetry of the parabola, and it can be used to immediately determine how many zeros it has.

The quadratic formula can be derived with a simple application of technique of completing the square. For this reason, the derivation is sometimes left as an exercise for students, who can thus experience rediscovery of this important formula. The explicit derivation is as follows.

Divide the quadratic equation by a, which is allowed because a is non-zero:

Subtract c/a from both sides of the equation, yielding:

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