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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