Quadratic RELATIONS
Perfect Square Trinomials
A trinomial that results from squaring a binomial is called a Perfect Square Trinomial. These trinomials can be factored by using patterns from expanding binomials.
Tips to remember!
1) The first and the last terms are perfect squares, meaning they can be square rooted
2) The middle term is twice the product of the square roots of the first and last terms
*If these points are not met, then the trinomial is not considered as a PERFECT SQUARE*
Examples:
a) (x+4)^2 b) (z+8)^2 c) (5d-4)^2
= (x+4)(x+4) = (z+8)(z+8) =(5d-4)(5d-4)
= x2+4x+4x+16 =z2+8x+8x+64 = 25d2-20d-20d+16
= x2+8x+16 =z2+16x+64 =25d2-40d+16
Now lets try!
Verify if these trinomials are perfect squares:
1) 25x2+30x+9 2) 5x2+16x+3 3) x2+8x+4
Answers:
1) 25x2+30x+9 2) 5x2+16x+3 3) x2+8x+4
- Perfect square - Not a perfect square - Perfect square
5 x 3= 15 x 2 = 30 1 x 4= 4 x 2 = 8
Difference of Squares
Difference of squares takes place when the middle terms in gone
For example: x2-36
[In this case the middle term is gone, however you can still factor]
Tip: If their an addition sign, it is not a difference of squares
Step 1: square root 1 and 36 if possible
1x and 6
Step 2: Place them into brackets to make them turn into factored form
(x+6) (x-6) [Rememeber when you are completing a difference of squares and placing them into brackets, put one bracket as postive and one as negative.
If we were to check: = (x+6) (x-6)
= x2-6x+6x+36
= x2+36
Example 2: 18x2-98b2
Step 1 : Common factor, since you can not square root right away
= 2(9x2-49b2)
Step 2: Square root 9x2 and 49b2 and place the terms into two binomials
= 2(3x-7b)(3x+7b)
Lets Practice:
1) y2-25 2) x2+56 3) 144y2-169
4) 12x2-27y2
Answers:
1) y2-25 2) x2+56 3) 144y2-169
= (y-5)(y+5) - not possible = (12y-13)(12y+13)
4) 12x2-27y2 [ Common factor ]
= 3(4x2-9y2)
= 3(2x-3y)(2x+3y)