Difference between revisions of "Quadratic equation"

(Copied From completing the Square for completeness.)
(General Solution For A Quadratic by Completing the Square)
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=== Quadratic Formula ===
 
=== Quadratic Formula ===
===General Solution For A Quadratic by Completing the Square===
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See [[Quadratic Formula]].
 
 
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 21:59, 17 June 2006

Quadratic Equations

A quadratic equation is an equation of form ${a}{x}^2+{b}{x}+{c}=0$. 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 $x^2-3x+2=0$ for x. Solution: $x^2-3x+2=0$ 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 $x^2-x-2x+2=0$ Next, we factor out our common terms to get: $x(x-1)-2(x-1)=0$. We can now factor the (x-1) term to get: $(x-1)(x-2)=0$. By a well know theorem, Either $(x-1)$ or $(x-2)$ 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

Completing the square

Quadratic Formula

See Quadratic Formula.