Are any of these perfect squares? No, these are perfect squares

 Recall: A perfect square is a number that is obtained by a product of the same number. ◦ EX. 16 and 25 are perfect squares, because 4 x 4 = 16 and 5 x 5 = 25  22 on the other hand is not a perfect square because 11 x 2 = 22  Now let’s look at what it means to be a “perfect square” in the context of a quadratic equation

 We agree that a number multiplied by itself will return a perfect square ◦ (5) x (5) = (5) 2 = 25  This is true for anything in the brackets ◦ EX. (☺) x (☺) = (☺) 2 ◦ EX. (♥) x (♥) = (♥) 2  So how do you think it is possible to state that the equation y = 4x x + 9 is a perfect square?

 y = 4x x + 9  Based on our previous conclusion, if we can write the expression as (something) 2, it is a perfect square  But what times itself gives 4x x + 9?

 In y = 4x x + 9, both the first and last numbers (4 and 9) are perfect squares ◦ 2 2 = 4 and 3 2 = 9 – we can use this  It turns out, that 4x x + 9 = (2x + 3) 2  This trick usually works, but expand the brackets to verify that

 Factor 25x 2 – 40x + 16  Using the trick we just found: ◦ 5 2 = 25, and 4 2 = 16, but here, the middle term is negative, so perhaps it is (5x – 4) 2  If you check this, you will see that it is correct ◦ (5x – 4)(5x – 4)  = 25x 2 – 20x – 20x + 16  = 25x 2 – 40x + 16

 Factor the following difference of squares: ◦ x 2 – 1 ◦ The coefficient in front of the x 2 is 1  1 x 1 = (1) 2 = 1  x 2 – 1 = (x + 1)(x – 1) ◦ So instead of being just equal to (x + 1) 2 or (x – 1) 2, it is equal to (x + 1)(x – 1) – this ensures that the middle term (with a single x) cancels out.

 A polynomial of the form a 2 + 2ab + b 2 or a 2 – 2ab + b 2 is a perfect square trinomial: ◦ a 2 + 2ab + b 2 can be factored as (a + b) 2 ◦ a 2 - 2ab + b 2 can be factored as (a - b) 2  A polynomial of the form a 2 – b 2 is a difference of squares and can be factored as (a + b)(a – b)