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Factoring Special Quadratics

Difference of Squares

1, 4, 9, 16, 25, 36... are all perfect squares.

 

When you are given 2 perfect square terms, there is an easy way to factor them:

->(a+b)(a-b)=a^2 - b^2

->To find a and b, square root the first and last term

 

Example:

 

16x^2 + 9

 

a=   16x^2  b=   9

 

 

= (4x + 9)(4x - 9)

 

->The last term in the first bracket is always positive, and in second bracket, it is always negative. Or vise-versa.

 

 

Perfect Square Trinomials

The trinomial that results from squaring a binomial is called a perfect square trinomial. They can be factored using the patterns from expanding binomials:

 

               a^2 + 2ab + b^2 = (a+b)^2

               a^2 - 2ab + b^2 = (a-b)^2

 

->The first and last terms are perfect squares

->The middle term is twice the product of the square root of the first term and the square root of the last term.

 

Example:

x^2 - 10x + 25                         ->Square root the 1st and last term 

=(x-5)^2                                  ->2ab is negative so the last term in the binomial squared will                                                   be negative

 

Example 2:

x^2 + 18x + 81

=(x+9)^2

 

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