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.

Answer:

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)$^{2}$

9th Grade Math

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