Square of Sum or Perfect Square Trinomials

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In some cases recognizing some common patterns in the quadratic equation will help you to factorize the quadratic.
In this lesson, we will look at the Square of Sum or Perfect Square Trinomials.

A Square of Sum is a type of quadratic equations of the form:

x² + 2bx + b² = (x + b)²

Example 1:	x2 + 2x + 1 = 0
	(x + 1)2 = 0
	x + 1 = 0 ⇒ x = -1
Example 2:	x2 + 6x + 9 = 0
	x2 + 2(3)x + 32 = 0
	(x + 3)2 = 0
	x + 3 = 0 ⇒ x = -3

Perfect Square Trinomials

We may check the pattern of the expression to determine whether it is a Perfect Square Trinomial, namely,

p² + 2pq + q² = ( p + q )² or

p²– 2 pq + q² = ( p – q )²

Example:

Factorize the following:

a) x² + 8x + 16

b) 4x²– 20x + 25

Solution:

a) x² + 8x + 16

= x² + 2(x)(4) + 4 2 ← Write in the form of p² + 2pq + q²

= (x + 4)²

b) 4x²– 20x + 25

= (2x)²– 2(2x)(5) + 5²← Write in the form of p² + 2pq + q²

= (2x – 5)²

How to factor perfect square trinomials?
If we have three terms with perfect square on both ends then try "perfect square trinomial"
Example:
Factor:
1. v² + 14v + 49
2. t² + 1/3 t + 1/36

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How to factor perfect square trinomials and the difference of two squares?
Example:
Factor each of the following.
a) y² + 10y + 25
b) 9x² - 12x + 4
c) 36m² - 84m + 49
d) 9k² + 10k + 4
e) y² - 25
f) 49h² - 36
g) 32w² - 18z²
h) 100q² + 29k²

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How to factor a perfect square trinomial?
Examples:
4x² - 12x + 9
16x² + 56x + 49
25² - 10x + 1
4x² + 16x + 16

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Factor Perfect Square Trinomials
Example:
x² + 12x + 36
x² - 16x + 64
4x² - 12x + 9
16x² + 40x + 25

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