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 - \(\frac{1}{z^{2}}\).
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