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

Question

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

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

Answer 1

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

 

Answer 2

 

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)