MM 1/2 Chapter 3 :: Quadratics

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3A :: Expanding and Collecting Like Terms

General binomial expansion: (a+b)(c+d) = a(c+d) + b(c+d)

= ac + ad + bc + bd

Perfect square expansion: (a+b)^2 = a^2 + 2ab + b^2m

Difference of two squares expansion: (a+b)(a-b) = a^2 - b^2

Example 1: Simplify 2(x-5)-3(x+5) by first expanding.

2(x-5)-3(x+5) = 2x - 10 - 3x - 15

= 2x - 3x - 10 - 15

= -x-25

Example 2: Expand 2x(3x-2)+3x(x-2)

2x(3x-2)+3x(x-2) = 6x^2 - 4x + 3x^2 - 6x

= 9x^2 - 10x

Example 3: Expand the following:

a. (x+3)(2x-3)

b. (x-3)(2x-2√2)

a. (x+3)(2x–3)

= x(2x-3)+3(2x-3)

= 2x^2 - 3x + 6x - 9

= 2x^2 + 3x - 9

b. (x-3)(2x-2√2)

= x(2x-2√2) - 3(2x-2√2)

= 2x^2 - 2√2x - 6x + 6√2

= 2x^2 - (2√2 + 6)x + 6√2

Example 4: Expand (2x-1)(3x^2+2x+4)

(2x - 1)(3x^2+2x+4) = 2x(3x^2+2x+4) - 1(3x^2+2x+4)

= 6x^2 + 4x^2 + 8x - 3x^2 - 2x - 4

= 6x^3 + x^2 + 6x -4

Example 5: Expand (3x-2)^2

(3x-2)^2 = (3x)^2 + 2(3x)(-2) + (-2)^2

= 9x^2 - 12x +4

Example 6: Expand:

a. (2x-4)(2x+4)

b. (x-2√7)(x+2√7)

a. (2x-4)(2x+4) = (2x)^2 - 4^2

= 4x^2 - 16

b. (x-2√7)(x+2√7) = x^2 - (2√7)^2

= x^2 - 28

Example 7: Expand (2a-b+c)(2a-b-c)

(2a-b+c)(2a-b-c) = ((2a - b) + c)((2a - b) - c)

= (2a - b)^2 - c^2

= 4a^2 - 4ab + b^2 - c^2

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