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Difference between revisions of "Quadratic equation"

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We can now factor the <math>(x-1)</math> term to get <math>(x-1)(x-2)=0</math>. By the zero-product property, either <math> (x-1) </math> or <math> (x-2) </math> equals zero.  
 
We can now factor the <math>(x-1)</math> term to get <math>(x-1)(x-2)=0</math>. By the zero-product property, either <math> (x-1) </math> or <math> (x-2) </math> equals zero.  
  
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We now have the pair of equations <math>x-1=0</math> and <math>x-2=0</math>. These give us the answers <math>x=1</math> and <math>x=2</math>, which can also be written as <math>x=\{1,\,2\}</math>. Plugging these back into the original equation, we find that both of these work! We are done.
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We now have the pair of equations <math>x-1=0</math> and <math>x-2=0</math>. These give us the answers <math>x=1</math> and <math>x=2</math>, which can also be written as <math>x=[1,\,2]</math>. 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 05:18, 18 December 2020

A quadratic equation in one variable is an equation of the form ${a}{x}^2+{b}{x}+{c}=0$, where $a$, $b$ and $c$ are constants (that is, they do not depend on $x$) and $x$ is the unknown variable. Quadratic equations are solved using one of three 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 $x$. Note: This is different for all quadratics; we cleverly chose this so that it has common factors.

Solution: $x^2-3x+2=0$

First, we expand the middle term: $x^2-x-2x+2=0$.

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 the zero-product property, either $(x-1)$ or $(x-2)$ equals zero.

We now have the pair of equations $x-1=0$ and $x-2=0$. These give us the answers $x=1$ and $x=2$, which can also be written as $x=[1,\,2]$. 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.

See Also

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