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Difference between revisions of "Quadratic equation"

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=== Quadratic Equations ===
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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]].
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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 ===
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The purpose of factoring is to turn a general quadratic into a product of [[binomial]]s. This is easier to illustrate than to describe.
  
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The purpose of factoring is to turn a general quadratic into a product of binomials. This is easier to illustrate than to describle
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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.
  
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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>
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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>
 
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Next, we factor out our common terms to get: <math>x(x-1)-2(x-1)=0</math>.
 
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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.
 
  
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=== Completing the square ===
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First, we expand the middle term: <math>x^2-x-2x+2=0</math>.
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[[ Completing the square ]]
 
  
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=== Quadratic Formula ===
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Next, we factor out our common terms to get <math>x(x-1)-2(x-1)=0</math>.
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===General Solution For A Quadratic by Completing the Square===
 
  
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Let the quadratic be in the form <math>a\cdot x^2+b\cdot x+c=0</math>.  
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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.  
  
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Moving c to the other side, we obtain
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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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<math>ax^2+bx=-c</math>
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=== Completing the square ===
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[[ Completing the square ]]
  
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Dividing by <math>{a}</math> and adding <math>\frac{b^2}{4a^2}</math> to both sides yields
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=== Quadratic Formula ===
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See [[Quadratic Formula]].
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<math>x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}</math>.
 
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Factoring the LHS gives
 
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<math>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}</math>
 
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As described above, an equation in this form can be solved, yielding
 
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<math>{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}</math>
 
  
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This formula is also called the [[Quadratic Formula]].
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== See Also ==
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* [[Discriminant]]
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* [[Vieta's Formulas]]
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* [[Quadratic Inequality]]
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* [[Factoring Quadratics]]
  
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We simply plug in a, b, and c and out pops the 2 values of x.
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[[Category:Algebra]]
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[[Category:Quadratic equations]]
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[[Category:Definition]]

Latest revision as of 12:04, 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

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