SUM AND DIFFERENCE OF SQUARES OF TWO BINOMIALS

Expand the square of the binomial (a + b)² using FOIL method :

(a + b)² = (a + b)(a + b)

(a + b)² = a ⋅ a + ab + ab + b ⋅ b

(a + b)² = a² + 2ab + b²

From the above working, it is clear that the expansion of (a + b)² is equal to sum of the squares of the two terms a and b and two times the product of the two terms a and b.

Expand the square of the binomial (a - b)² using FOIL method :

(a - b)² = (a - b)(a - b)

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

(a - b)² = a² - ab - ab - b²

(a - b)² = a² - 2ab + b²

The following expansions of the squares of two binomials are considered to be algebraic identities.

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

It will be helpful to remember the above two algebraic identities to solve problems on expanding the square of a binomial.

Sum of the Squares of Two Binomials

(a + b)² + (a - b)² :

= (a² + 2ab + b²) + (a² - 2ab + b²)

= a² + 2ab + b² + a² - 2ab + b²

Combine the like terms by grouping.

= (a² + a²) + (2ab - 2ab) + (b² + b²)

= 2a² + 0 + 2b²

= 2a² + 2b²

= 2(a² + b²)

Therefore,

(a + b)² + (a - b)² = 2(a² + b²)

Difference of the Squares of Two Binomials

(a + b)² - (a - b)² :

= (a² + 2ab + b²) - (a² - 2ab + b²)

= a² + 2ab + b² - a² + 2ab - b²

Combine the like terms by grouping.

= (a² - a²) + (2ab + 2ab) + (b² - b²)

= 0 + 2ab + 0

= 2ab

Therefore,

(a + b)² - (a - b)² = 2ab

Solved Problems

Example 1 :

Simplify :

(2x + 3y)² + (2x - 3y)²

Solution :

Using (a + b)² + (a - b)² = 2(a² + b²),

(2x + 3y)² + (2x - 3y)² = 2[(2x)² + (3y)²]

= 2(2²x² + 3²y²)

= 2(4x² + 9y²)

= 8x² + 18y²

Example 2 :

Simplify :

(2x + 3y)² - (2x - 3y)²

Solution :

Using (a + b)² - (a - b)² = 2ab,

(2x + 3y)² - (2x - 3y)² = 2(2x)(3y)

= 12xy

Example 3 :

Evaluate :

101² + 99²

Solution :

101² + 99² = (100 + 1)² + (100 - 1)²

Using (a + b)² + (a - b)² = 2(a² + b²),

= 2(100² + 1²)

= 2(10000 + 1)

= 2(10001)

= 20002

Example 4 :

Evaluate :

101² - 99²

Solution :

101² - 99² = (100 + 1)² - (100 - 1)²

Using (a + b)² - (a - b)² = 2ab,

= 2(100)(1)

= 200

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