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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