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Difference between revisions of "Perfect square"
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An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect squares are 0, 1, 4, 9, 16, 25, 36.
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An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect squares are <math>0, 1, 4, 9, 16, 25, 36</math>.
  
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The sum of the first <math>n</math> square numbers (not including 0) is <math>\frac{n(n+1)(2n+1)}{6}</math>
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The sum of the first <math>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</math>
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An integer <math>n</math> is a perfect square [[iff]] it is a [[quadratic residue]] [[modulo]] all but finitely [[prime]]s.
  
 
== Perfect Square Trinomials ==
 
== Perfect Square Trinomials ==
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A type of perfect square is an equation that is a perfect square trinomial. For example, <math>(x+a)^2=x^2+2xa+a^2</math>.
  
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Another type of perfect square is an equation that is a perfect square trinomial. Take for example
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Perfect square trinomials are a type of quadratic equation that have <math>3</math> terms and contain <math>1</math> unique root.
  
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<math>(x+a)^2=x^2+2xa+a^2</math>.
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For any quadratic equation in the form <math>ax^2+bx+c</math>, it is a perfect square trinomial iff <math>b=2\sqrt{ac}</math>.
  
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Perfect square trinomials are a type of quadratic equation that have 3 terms and contain 1 unique root.
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==See also ==
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* [[Perfect cube]]
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* [[Perfect power]]
  
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For any quadratic equation in the form <math>ax^2+bx+c</math>, it is a perfect square trinomial [[iff]] <math>b=a\sqrt{c}</math>.
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{{stub}}
  
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[[Category:Definition]]

Revision as of 13:41, 9 March 2013

An integer $n$ is said to be a perfect square if there is an integer $m$ so that $m^2=n$. The first few perfect squares are $0, 1, 4, 9, 16, 25, 36$.

The sum of the first $n$ square numbers (starting with $1$) is $\frac{n(n+1)(2n+1)}{6}$

An integer $n$ is a perfect square iff it is a quadratic residue modulo all but finitely primes.

Perfect Square Trinomials

A type of perfect square is an equation that is a perfect square trinomial. For example, $(x+a)^2=x^2+2xa+a^2$.

Perfect square trinomials are a type of quadratic equation that have $3$ terms and contain $1$ unique root.

For any quadratic equation in the form $ax^2+bx+c$, it is a perfect square trinomial iff $b=2\sqrt{ac}$.

See also

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