Expansion using Special Algebraic Identities
Now we will learn how to expand special products of algebraic expressions:
Perfect Squares and Difference of Two Squares
Perfect squares:
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
Difference of two squares:
(a + b)(a - b) = a² - b²
- Perfect Squares Example:
a) (x + 5)² = x² + 2(x)(5) + 5² <------ Use (a + b)² = a² + 2ab +b²
= x² + 10x + 25 Here a = x and b = 5.
b) (3x - 4)² = (3x)² - 2(3x)(4) + 4² <------ Use (a - b)² = a² - 2ab + b²
= 9x² - 24x + 16
*Be careful when expanding expressions like (x + 5)².
(x + 5)² ≠ x² + 25
(x + 5)² = (x + 5)(x + 5)
- Difference of two Squares:
a) (x+1)(x-1) = x² - 1² <------ Use (a + b)(a - b) = a² - b²
= x² - 1
b) (x + y + 2)(x + y - 2)
= [(x + y) + 2)][(x + y) - 2] <------- Use (a+b)(a-b) = a² - b²
= (x + y)² - 2² Here a = (x + y) and b = 2
= x² + 2xy + y² - 4 <------- Use (a+b)² = a² + 2ab + b².
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