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🔢 Chapter 4: Exploring Algebraic Identities • Exercise Set 4.1

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Exercise Set 4.1 2Exercise Set 4.2 2Exercise Set 4.3 1Exercise Set 4.4 2Exercise Set 4.5 1End-of-Chapter Exercises 6All Exercises 14
Q1

Exercise Set 4.1, Q1: Using the identity $(a+b)^2 = a^2 + 2ab + b^2$, expand the following:
(i) $(7x+4y)^2$
(ii) $(\frac{7}{5}x + \frac{3}{2}y)^2$
(iii) $(2.5p + 1.5q)^2$
(iv) $(\frac{3}{4}s + 8t)^2$
(v) $(x + \frac{1}{2y})^2$
(vi) $(\frac{1}{x} + \frac{1}{y})^2$

Answer:
(i) $(7x)^2 + 2(7x)(4y) + (4y)^2 = \mathbf{49x^2 + 56xy + 16y^2}$
(ii) $(\frac{7}{5}x)^2 + 2(\frac{7}{5}x)(\frac{3}{2}y) + (\frac{3}{2}y)^2 = \mathbf{\frac{49}{25}x^2 + \frac{21}{5}xy + \frac{9}{4}y^2}$
(iii) $(2.5p)^2 + 2(2.5p)(1.5q) + (1.5q)^2 = \mathbf{6.25p^2 + 7.5pq + 2.25q^2}$
(iv) $(\frac{3}{4}s)^2 + 2(\frac{3}{4}s)(8t) + (8t)^2 = \mathbf{\frac{9}{16}s^2 + 12st + 64t^2}$
(v) $x^2 + 2(x)(\frac{1}{2y}) + (\frac{1}{2y})^2 = \mathbf{x^2 + \frac{x}{y} + \frac{1}{4y^2}}$
(vi) $(\frac{1}{x})^2 + 2(\frac{1}{x})(\frac{1}{y}) + (\frac{1}{y})^2 = \mathbf{\frac{1}{x^2} + \frac{2}{xy} + \frac{1}{y^2}}$

Q2

Exercise Set 4.1, Q2: Using the same identity, find the values of the following:
(i) $(64)^2$
(ii) $(105)^2$
(iii) $(205)^2$

Answer:
(i) $(60 + 4)^2 = 60^2 + 2(60)(4) + 4^2 = 3600 + 480 + 16 = \mathbf{4096}$
(ii) $(100 + 5)^2 = 100^2 + 2(100)(5) + 5^2 = 10000 + 1000 + 25 = \mathbf{11025}$
(iii) $(200 + 5)^2 = 200^2 + 2(200)(5) + 5^2 = 40000 + 2000 + 25 = \mathbf{42025}$