# What are these formulas used for and how can you tell when to apply it?

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(a+b)^2= a^2+2ab+b^2

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

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Difference of two squares formula

a2 - b2 = (a + b) (a - b)

This formula is used to factorize some special algebraic expressions.

For example:

1. x2 - 16 = x2 - 42

= (x + 4 ) ( x - 4)

2. 16x6 - y8 = (4x3)2 - (y4)2

= (4x3 + y4) (4x3 - y4)

The difference of two squares can be used to find the linear factors of the sum of two squares, using complex number coefficients.

For example:

1. a2 + 5

= a2 - (- 5)

= a2 - (√-5)2

Substitute imaginary unit value i2 = - 1

a2 - (√i25)2

a2 - (i√  5)2

Apply the difference of two squares formula

= (a + i√5)   (a - i√5)

The lineae factors are (a + i√5)   (a - i√5).

The difference of two squares can also be used as a arithmetical short cut.

For example Multiply 101 . 99 without pencil and paper

write the expression as (100 + 1) ( 100 - 1)

Apply formula a2 - b2 = (a + b) (a - b)

(100 + 1) ( 100 - 1) = 1002 - 12 = 10000 - 1 = 9999

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Formula for short multiplication: The Square of a sum

For example consider (a + b) (m + n) can be multiplied using FOIL (First Outer Inner Last) method.

(a + b) (m + n) = am + an + bm + bn

Now consider the case when the letters inside the parentheses are the same

(a + b) (a + b) = aa + ab + ba + bb = a2 + 2ab + b2

(a + b)2 = a2 + 2ab + b2

Examples:

Factor 4x4 + 625y4.

The square of a sum formula (a + b)2 = a2 + 2ab + b2

Rewrite the formula as   a2 +   b2 = (a + b)2 - 2ab

a2 +   b2 = (a + b) (a + b) - 2ab = (a + b + √2ab) (a + b - √2ab)

Let A  =  2x ² and B  =  25y ²; then 2AB  =  100x ²y ² is a perfect square and √(2AB)  =  10xy.

4x4 + 625y4 = (2x ² + 25y ² + 10xy) (2x ² + 25y ² − 10xy)