Quadratic equation

A quadratic equation can take two forms. The general formula, written as a function of <math>x</math>, <math>y=f(x)</math>, is:

<math>y = ax^2 + bx + c</math>

The graph of a quadratic equation is a parabola, one of the conic sections. (In some cases, the parabola collapses, most obviously when <math>a = 0</math>)

The points where this curve crosses the x axis are represented by the second form of the equation:

<math>ax^2 + bx + c = 0</math>

These are solved for using the quadratic formula, which will not only solve for real roots, but result in the imaginary roots if the parabola does not actually cross the y axis (this is when <math>4ac</math> is greater than <math>b^2</math>).

Quadratic equations are very important in calculating the motion of bodies under constant acceleration, i.e., gravity (when close to the earth's surface).

The derivative of a quadratic equation is a simple linear function:

<math>\frac{dy}{dx} = 2ax + b</math>

Factoring

Quadratic equations can be simplified by factoring it into <math>(x + r)(x + s)</math>, where <math>r + s</math> equals B in <math>y = x^2 + bx + c</math> and <math>rs</math> equals <math>c</math> in the above equation. Equations that cannot be easily factored this way can become easy to factor by completing the square.

If the coefficient of X is not one

If the coefficient of X is not 1, then one can turn <math>ax^2 + bx</math> into <math>a(x^2 + b/ax)</math>. This works because both <math>x^2</math> and <math>bx</math> were divided by <math>a</math>, which was put back later via multiplication.