Simplification of (a + b)(a – b)

We will discuss here about the Simplification of (a + b)(a – b).

(a + b)(a – b) = a(a – b) + b(a – b)

                     = a\(^{2}\) - ab + ba - b\(^{2}\)

                     = a\(^{2}\) - b\(^{2}\)

Thus, we have (a + b)(a - b) = a\(^{2}\) - b\(^{2}\)

Solved Examples on Simplification of (a + b)(a – b)

1. Simplify: (3m – 4n + 2)(3m – 4n – 2)

Solution:

Given expression = (3m – 4n + 2)(3m – 4n – 2)

                         = [(3m – 4n) + 2][(3m – 4n) – 2]

Let 3m – 4n = x. Then,

Given expression = (x + 2)(x – 2)

                          = x\(^{2}\) – 2\(^{2}\)

                          = x\(^{2}\) – 4

= (3m – 4n)\(^{2}\) – 4, [plug-in x = 3m – 4n]

= (3m)\(^{2}\) – 2 ∙ 3m ∙ 4n + (4n)\(^{2}\) - 4

= 9m\(^{2}\) – 24mn + 16n\(^{2}\) – 4.

 

2. Simplify: (z - \(\frac{1}{z}\) + 3)(z + \(\frac{1}{z}\) + 3)

Solution:

Given expression = (z - \(\frac{1}{z}\) + 3)(z + \(\frac{1}{z}\) + 3)

                         = [(z + 3) - \(\frac{1}{z}\)][(z + 3) + \(\frac{1}{z}\)]

Let z + 3 = k. Then,

Given expression = (k - \(\frac{1}{z}\))(k + \(\frac{1}{z}\))

                         = k\(^{2}\) – (\(\frac{1}{z}\))\(^{2}\)

                         = (z + 3)\(^{2}\) – (\(\frac{1}{z}\))\(^{2}\), [plug-in k = z + 3]

                         = z\(^{2}\) + 2 ∙ z ∙ 3 + 3\(^{2}\) - \(\frac{1}{z^{2}}\)

                         = z\(^{2}\) + 6z + 9 - \(\frac{1}{z^{2}}\).

9th Grade Math

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