# 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 - $$\frac{1}{z^{2}}$$.

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