Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Quadratic equation"

(Copied From completing the Square for completeness.)
 
(18 intermediate revisions by 12 users not shown)
Line 1: Line 1:
−
=== Quadratic Equations ===
+
A '''quadratic equation''' in one [[variable]] is an [[equation]] of the form <math> {a}{x}^2+{b}{x}+{c}=0</math>, where <math>a</math>, <math>b</math> and <math>c</math> are [[constant]]s (that is, they do not depend on <math>x</math>) and <math>x</math> is the unknown variable. Quadratic equations are solved using one of three main strategies: [[factoring]], [[completing the square]] and the [[quadratic formula]].
−
 
 
−
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 [[binomial]]s. This is easier to illustrate than to describe.
  
−
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 <math>x^2-3x+2=0</math> for <math>x</math>. Note: This is different for all quadratics; we cleverly chose this so that it has common factors.
  
−
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>
 
−
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, Either <math> (x-1) </math> or <math> (x-2) </math> 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 ===
+
First, we expand the middle term: <math>x^2-x-2x+2=0</math>.
−
[[ Completing the square ]]
 
  
−
=== Quadratic Formula ===
+
Next, we factor out our common terms to get <math>x(x-1)-2(x-1)=0</math>.
−
===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>.  
+
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.  
  
−
Moving c to the other side, we obtain
+
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.
  
−
<math>ax^2+bx=-c</math>
+
=== Completing the square ===
 +
[[ Completing the square ]]
  
−
Dividing by <math>{a}</math> and adding <math>\frac{b^2}{4a^2}</math> to both sides yields
+
=== Quadratic Formula ===
−
 
+
See [[Quadratic Formula]].
−
<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]].
+
== See Also ==
 +
* [[Discriminant]]
 +
* [[Vieta's Formulas]]
 +
* [[Quadratic Inequality]]
 +
* [[Factoring Quadratics]]
  
−
We simply plug in a, b, and c and out pops the 2 values of x.
+
[[Category:Algebra]]
 +
[[Category:Quadratic equations]]
 +
[[Category:Definition]]

Latest revision as of 11:04, 15 July 2021

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Quadratic_equation&oldid=158541"