Difference between revisions of "Quadratic equation"

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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.
 +
 
Solution: <math>x^2-3x+2=0</math>
 
Solution: <math>x^2-3x+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>.
 
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 known 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.
+
We can now factor the (x-1) term to get: <math>(x-1)(x-2)=0</math>. By a well known 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 14:23, 27 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 describe.

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 known theorem, either $(x-1)$ or $(x-2)$ 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

Quadratic Formula

See Quadratic Formula.