Difference between revisions of "Quadratic equation"
(fixed link) |
IntrepidMath (talk | contribs) (Copied From completing the Square for completeness.) |
||
Line 17: | Line 17: | ||
=== Quadratic Formula === | === Quadratic Formula === | ||
+ | ===General Solution For A Quadratic by Completing the Square=== | ||
+ | |||
+ | Let the quadratic be in the form <math>a\cdot x^2+b\cdot x+c=0</math>. | ||
+ | |||
+ | Moving c to the other side, we obtain | ||
+ | |||
+ | <math>ax^2+bx=-c</math> | ||
+ | |||
+ | Dividing by <math>{a}</math> and adding <math>\frac{b^2}{4a^2}</math> to both sides yields | ||
+ | |||
+ | <math>x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}</math>. | ||
+ | |||
+ | Factoring the LHS gives | ||
+ | |||
+ | <math>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}</math> | ||
+ | |||
+ | As described above, an equation in this form can be solved, yielding | ||
+ | |||
+ | <math>{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}</math> | ||
+ | |||
+ | This formula is also called the [[Quadratic Formula]]. | ||
+ | |||
+ | We simply plug in a, b, and c and out pops the 2 values of x. |
Revision as of 20:55, 17 June 2006
Contents
[hide]Quadratic Equations
A quadratic equation is an equation of form . a, b, and c are constants, and x is the unknown variable. Quadratic Equations are solved using 3 main strategies, factoring, completing the square, and the quadratic formula.
Factoring
The purpose of factoring is to turn a general quadratic into a product of binomials. This is easier to illustrate than to describle
Example: Solve the equation for x. Solution: First we expand the middle term. This is different for all quadratics. We cleverly choose this so that it has common factors. We now have Next, we factor out our common terms to get: . We can now factor the (x-1) term to get: . By a well know theorem, Either or equals zero. We now have the pair or equations x-1=0, or x-2=0. These give us answers of x=1 or x-2. Plugging these back into the original equation, we find that both of these work! We are done.
Completing the square
Quadratic Formula
General Solution For A Quadratic by Completing the Square
Let the quadratic be in the form .
Moving c to the other side, we obtain
Dividing by and adding to both sides yields
.
Factoring the LHS gives
As described above, an equation in this form can be solved, yielding
This formula is also called the Quadratic Formula.
We simply plug in a, b, and c and out pops the 2 values of x.