# Worksheet on Simplification of (a + b)(a – b)

Practice the questions given in the worksheet on simplification of (a + b)(a – b).

1. Simplify by applying standard formula.

(i) (5x – 9)(5x + 9)

(ii) (2x + 3y)(2x – 3y)

(iii) (a + b – c)(a – b + c)

(iv) (x + y – 3)(x + y + 3)

(v) (1 + a)(1 – a)(1 + a$$^{2}$$)

[Hint: Given expression = (1 - a$$^{2}$$)(1 + a$$^{2}$$) = 1 -(a$$^{2}$$)$$^{2}$$.]

(vi) (a + $$\frac{2}{a}$$ – 1)(a - $$\frac{2}{a}$$ – 1)

2. If a - $$\frac{1}{a}$$ = 3, find the value of a$$^{2}$$ - $$\frac{1}{a^{2}}$$.

[Hint: (a + $$\frac{1}{a}$$)$$^{2}$$ = (a - $$\frac{1}{a}$$)$$^{2}$$ + 4a ∙ $$\frac{1}{a}$$ = 3$$^{2}$$ + 4 = 13.

Therefore, a + $$\frac{1}{a}$$ = ±$$\sqrt{13}$$.

Now (a + $$\frac{1}{a}$$)(a - $$\frac{1}{a}$$) = ±$$\sqrt{13}$$ × 3 = ±3$$\sqrt{13}$$]

3. If x - $$\frac{1}{x}$$ = $$\frac{3}{2}$$, find the value of

(i) x + $$\frac{1}{x}$$

(ii) x$$^{2}$$ + $$\frac{1}{x^{2}}$$

(iii) x$$^{2}$$ - $$\frac{1}{x^{2}}$$

(iv) x$$^{4}$$ + $$\frac{1}{x^{4}}$$

(v) x$$^{4}$$ - $$\frac{1}{x^{4}}$$

4. (i) Simplify: (1 – x)(1 + x)(1 + x$$^{2}$$)(1 + x$$^{4}$$).

[Hint: Given expression = (1 - x$$^{2}$$)(1 + x$$^{2}$$)(1 + x$$^{4}$$)

= (1 - x$$^{4}$$)(1 + x$$^{4}$$)

= 1 - (x$$^{4}$$)$$^{2}$$

= 1 - x$$^{8}$$]

(ii) Express: (x$$^{2}$$ + 5x + 12)(x$$^{2}$$ – 5x + 12) as a difference of two squares.

(iii) If $$\frac{a}{b}$$ = $$\frac{b}{c}$$, prove that (a + b + c)(a – b + c) = a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$.

[Hint: (a + b + c)(a – b + c) = {(a + c) + b}{(a + c) - b)}

= (a + c)$$^{2}$$ - b$$^{2}$$

= a$$^{2}$$ + 2ac + c$$^{2}$$ - b$$^{2}$$

= a$$^{2}$$ + 2b$$^{2}$$ + c$$^{2}$$ - b$$^{2}$$

(Since, $$\frac{a}{b}$$ = $$\frac{b}{c}$$ implies = ac = b$$^{2}$$)]

Answers for the worksheet on simplification of (a + b)(a – b) are given below.

1. (i) 25x$$^{2}$$ - 81

(ii) 4x$$^{2}$$ – 9y$$^{2}$$

(iii) a$$^{2}$$ – b$$^{2}$$ – c$$^{2}$$ + 2bc

(iv) x$$^{2}$$ + 2xy + y$$^{2}$$ - 9

(v) 1 – a$$^{4}$$

(vi) a$$^{2}$$ – 2a + 1 - $$\frac{4}{a^{2}}$$

2. ± 3$$\sqrt{3}$$

3. (i) ±$$\frac{5}{2}$$

(ii) $$\frac{17}{4}$$

(iii) ±$$\frac{15}{4}$$

(iv) $$\frac{257}{16}$$

(v) ±$$\frac{255}{16}$$

4. (i) 1 - x$$^{8}$$

(ii) (x$$^{2}$$ + 12)$$^{2}$$ – (5x)$$^{2}$$

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