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)\(^{2}\)
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