Difference between revisions of "Quadratic equation"
(→General Solution For A Quadratic by Completing the Square) |
m (proofreading) |
||
| Line 1: | Line 1: | ||
=== Quadratic Equations === | === Quadratic Equations === | ||
| − | A quadratic equation is an equation of form <math> {a}{x}^2+{b}{x}+{c}=0</math>. a, b, and c are constants, and x is the unknown variable. Quadratic Equations are solved using 3 main strategies | + | A quadratic equation is an equation of form <math> {a}{x}^2+{b}{x}+{c}=0</math>. 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 === | === Factoring === | ||
| − | The purpose of factoring is to turn a general quadratic into a product of binomials. This is easier to illustrate than to | + | The purpose of factoring is to turn a general quadratic into a product of binomials. This is easier to illustrate than to describe. |
Example: Solve the equation <math>x^2-3x+2=0</math> for x. | Example: Solve the equation <math>x^2-3x+2=0</math> for x. | ||
| Line 11: | Line 11: | ||
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 <math>x^2-x-2x+2=0</math> | 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 <math>x^2-x-2x+2=0</math> | ||
Next, we factor out our common terms to get: <math>x(x-1)-2(x-1)=0</math>. | Next, we factor out our common terms to get: <math>x(x-1)-2(x-1)=0</math>. | ||
| − | We can now factor the (x-1) term to get: <math>(x-1)(x-2)=0</math>. By a well know theorem, | + | We can now factor the (x-1) term to get: <math>(x-1)(x-2)=0</math>. By a well know theorem, either <math> (x-1) </math> or <math> (x-2) </math> equals zero. We now have the pair of 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 === | ||
Revision as of 11:55, 19 June 2006
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 describe.
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 of 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
See Quadratic Formula.
