Difference between revisions of "Quadratic equation"
(Added information about if the coefficient of X is not one) |
(→Factoring: made completing the square a wikilink) |
||
| Line 19: | Line 19: | ||
== Factoring == | == 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.<ref>http://www.regentsprep.org/regents/math/algtrig/ate3/quadlesson2.htm</ref> Equations that cannot be easily factored this way can become easy to factor by completing the square. | + | 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.<ref>http://www.regentsprep.org/regents/math/algtrig/ate3/quadlesson2.htm</ref> 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 one === | ||
Revision as of 06:19, December 23, 2014
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.[1] 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.