Mathematics Solutions Solutions for Class 8 Math Chapter 5 - Expansion Formulae

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Mathematics Solutions Solutions for Class 8 Math Chapter 5 Expansion Formulae are provided here with simple step-by-step explanations. These solutions for Expansion Formulae are extremely popular among class 8 students for Math Expansion Formulae Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Mathematics Solutions Book of class 8 Math Chapter 5 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Mathematics Solutions Solutions. All Mathematics Solutions Solutions for class 8 Math are prepared by experts and are 100% accurate.

Page No 24:

Question 1:

Expand
(1) (a+2)(a−1)
(2) (m−4)(m+6)
(3) (p+8)(p−3)
(4) (13+x)(13−x)
(5) (3x+4y)(3x+5y)
(6) (9x−5t)(9x+3t)
(7) $(m + \frac{2}{3})$ $(m - \frac{7}{3})$
(8) $(x + \frac{1}{x})$ $(x - \frac{1}{x})$
(9) $(\frac{1}{y} + 4)$ $(\frac{1}{y} - 9)$

Answer:

It is known that,
$(x + a) (x + b) = x^{2} + (a + b) x + a b$
$(1) (a + 2) (a - 1) = {(a)}^{2} + \{2 + (- 1)\} a + (2) \times (- 1) = a^{2} + (2 - 1) a - 2 = a^{2} + a - 2$
$(2) (m - 4) (m + 6) = {(m)}^{2} + \{(- 4) + 6\} m + (- 4) \times (6) = m^{2} + (- 4 + 6) m - 24 = m^{2} + 2 m - 24$
$(3) (p + 8) (p - 3) = {(p)}^{2} + \{8 + (- 3)\} p + (8) \times (- 3) = p^{2} + (8 - 3) p - 24 = p^{2} + 5 p - 24$
$(4) (13 + x) (13 - x) = - [(x + 13) (x - 13)] = - [{(x)}^{2} + \{13 + (- 13)\} x + (13) \times (- 13)] = - [x^{2} + (13 - 13) x - 169] = - [x^{2} - 169] = 169 - x^{2}$
$(5) (3 x + 4 y) (3 x + 5 y) = {(3 x)}^{2} + \{4 y + 5 y\} (3 x) + (4 y) \times (5 y) = 9 x^{2} + 9 y \times 3 x + 20 y^{2} = 9 x^{2} + 27 x y + 20 y^{2}$
$(6) (9 x - 5 t) (9 x + 3 t) = {(9 x)}^{2} + \{(- 5 t) + (3 t)\} (9 x) + (- 5 t) \times (3 t) = 81 x^{2} + (- 2 t) (9 x) - 15 t^{2} = 81 x^{2} - 18 x t - 15 t^{2}$
$(7) (m + \frac{2}{3}) (m - \frac{7}{3}) = {(m)}^{2} + \{\frac{2}{3} + (- \frac{7}{3})\} m + (\frac{2}{3}) \times (- \frac{7}{3}) = m^{2} + (\frac{2}{3} - \frac{7}{3}) m - \frac{14}{9} = m^{2} - \frac{5 m}{3} - \frac{14}{9}$
$(8) (x + \frac{1}{x}) (x - \frac{1}{x}) = {(x)}^{2} + \{\frac{1}{x} + (- \frac{1}{x})\} x + (\frac{1}{x}) \times (- \frac{1}{x}) = x^{2} + (\frac{1}{x} - \frac{1}{x}) x - \frac{1}{x^{2}} = x^{2} - \frac{1}{x^{2}}$
$(9) (\frac{1}{y} + 4) (\frac{1}{y} - 9) = {(\frac{1}{y})}^{2} + \{4 + (- 9)\} \frac{1}{y} + (4) \times (- 9) = \frac{1}{y^{2}} + (- 5) \times \frac{1}{y} - 36 = \frac{1}{y^{2}} - \frac{5}{y} - 36$

Page No 25:

Question 1:

Expand
(1) ${(k + 4)}^{3}$
(2) ${(7 x + 8 y)}^{3}$
(3) ${(7 + m)}^{3}$
(4) ${(52)}^{3}$
(5) ${(101)}^{3}$
(6) ${(x + \frac{1}{x})}^{3}$
(7) ${(2 m + \frac{1}{5})}^{3}$
(8) ${(\frac{5 x}{y} + \frac{y}{5 x})}^{3}$

Answer:

It is known that,
${(a + b)}^{3} = a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$
$(1) {(k + 4)}^{3} = {(k)}^{3} + {(4)}^{3} + 3 \times {(k)}^{2} \times (4) + 3 \times (k) \times {(4)}^{2} = k^{3} + 64 + 12 k^{2} + 48 k$
$(2) {(7 x + 8 y)}^{3} = {(7 x)}^{3} + {(8 y)}^{3} + 3 \times {(7 x)}^{2} \times (8 y) + 3 \times (7 x) \times {(8 y)}^{2} = 343 x^{3} + 512 y^{3} + 1176 x^{2} y + 1344 x y^{2}$
$(3) {(7 + m)}^{3} = {(7)}^{3} + {(m)}^{3} + 3 \times {(7)}^{2} \times (m) + 3 \times (7) \times {(m)}^{2} = 343 + m^{3} + 147 m + 21 m^{2}$
$(4) {(52)}^{3} = {(50 + 2)}^{3} = {(50)}^{3} + {(2)}^{3} + 3 \times {(50)}^{2} \times (2) + 3 \times (50) \times {(2)}^{2} = 125000 + 8 + 15000 + 600 = 140608$
$(5) {(101)}^{3} = {(100 + 1)}^{3} = {(100)}^{3} + {(1)}^{3} + 3 \times {(100)}^{2} \times 1 + 3 \times 100 \times {(1)}^{2} = 1000000 + 1 + 30000 + 300 = 1030301$
$(6) {(x + \frac{1}{x})}^{3} = {(x)}^{3} + {(\frac{1}{x})}^{3} + 3 \times {(x)}^{2} \times (\frac{1}{x}) + 3 \times (x) \times {(\frac{1}{x})}^{2} = x^{3} + \frac{1}{x^{3}} + 3 x + \frac{3}{x}$
$(7) {(2 m + \frac{1}{5})}^{3} = {(2 m)}^{3} + {(\frac{1}{5})}^{3} + 3 \times {(2 m)}^{2} \times (\frac{1}{5}) + 3 \times (2 m) \times {(\frac{1}{5})}^{2} = 8 m^{3} + \frac{1}{125} + \frac{12 m^{2}}{5} + \frac{6 m}{25}$
$(8) {(\frac{5 x}{y} + \frac{y}{5 x})}^{3} = {(\frac{5 x}{y})}^{3} + {(\frac{y}{5 x})}^{3} + 3 \times {(\frac{5 x}{y})}^{2} \times (\frac{y}{5 x}) + 3 \times (\frac{5 x}{y}) \times {(\frac{y}{5 x})}^{2} = \frac{125 x^{3}}{y^{3}} + \frac{y^{3}}{125 x^{3}} + \frac{15 x}{y} + \frac{3 y}{5 x}$

Page No 27:

Question 1:

Expand
(1) (2m − 5)³
(2) (4 − p)³
(3) (7x − 9y)³
(4) (58)³
(5) (198)³
(6) ${(2 p - \frac{1}{2 p})}^{3}$
(7) ${(1 - \frac{1}{a})}^{3}$
(8) ${(\frac{x}{3} - \frac{3}{x})}^{3}$

Answer:

It is known that,
${(a - b)}^{3} = a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2}$
$(1) {(2 m - 5)}^{3} = {(2 m)}^{3} - {(5)}^{3} - 3 \times {(2 m)}^{2} \times 5 + 3 \times 2 m \times {(5)}^{2} = 8 m^{3} - 125 - 60 m^{2} + 150 m$
$(2) {(4 - p)}^{3} = {(4)}^{3} - {(p)}^{3} - 3 \times {(4)}^{2} \times (p) + 3 \times 4 \times {(p)}^{2} = 64 - p^{3} - 48 p + 12 p^{2}$
$(3) {(7 x - 9 y)}^{3} = {(7 x)}^{3} - {(9 y)}^{3} - 3 \times {(7 x)}^{2} \times 9 y + 3 \times 7 x \times {(9 y)}^{2} = 343 x^{3} - 729 y^{2} - 1323 x^{2} y + 1701 x y^{2}$
$(4) {(58)}^{3} = {(60 - 2)}^{3} = {(60)}^{3} - {(2)}^{3} - 3 \times {(60)}^{2} \times (2) + 3 \times 60 \times {(2)}^{2} = 216000 - 8 - 21600 + 720 = 195112$
$(5) {(198)}^{3} = {(200 - 2)}^{3} = {(200)}^{3} - {(2)}^{3} - 3 \times {(200)}^{2} \times 2 + 3 \times 200 \times {(3)}^{2} = 8000000 - 8 - 240000 + 5400 = 7765392$
$(6) {(2 p - \frac{1}{2 p})}^{3} = {(2 p)}^{3} - {(\frac{1}{2 p})}^{3} - 3 \times {(2 p)}^{2} \times \frac{1}{2 p} + 3 \times 2 p \times {(\frac{1}{2 p})}^{2} = 8 p^{3} - \frac{1}{8 p^{3}} - 6 p + \frac{3}{2 p}$
$(7) {(1 - \frac{1}{a})}^{3} = {(1)}^{3} - {(\frac{1}{a})}^{3} - 3 \times {(1)}^{2} \times \frac{1}{a} + 3 \times 1 \times {(\frac{1}{a})}^{2} = 1 - \frac{1}{a^{3}} - \frac{3}{a} + \frac{3}{a^{2}}$
$(8) {(\frac{x}{3} - \frac{3}{x})}^{3} = {(\frac{x}{3})}^{3} - {(\frac{3}{x})}^{3} - 3 \times {(\frac{x}{3})}^{2} \times (\frac{3}{x}) + 3 \times (\frac{x}{3}) \times {(\frac{3}{x})}^{2} = \frac{x^{3}}{27} - \frac{27}{x^{3}} - x + \frac{9}{x}$

Page No 27:

Question 2:

Simplify
(1) (2a + b)³− (2a − b)³
(2) (3r − 2k)³ + (3r + 2k)³
(3) (4a − 3)³− (4a +3)³
(4) (5x − 7y)³+ (5x + 7y)³

Answer:

It is known that,
${(a + b)}^{3} = a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}; {(a - b)}^{3} = a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2}$
$(1) {(2 a + b)}^{3} - {(2 a - b)}^{3} = {(2 a)}^{3} + {(b)}^{3} + 3 \times {(2 a)}^{2} \times b + 3 \times 2 a \times {(b)}^{2} - \{{(2 a)}^{3} - {(b)}^{3} - 3 \times {(2 a)}^{2} \times b + 3 \times 2 a \times {(b)}^{2}\} = 8 a^{3} + b^{3} + 12 a^{2} b + 6 a b^{2} - 8 a^{3} + b^{3} + 12 a^{2} b - 6 a b^{2} = 2 b^{3} + 24 a^{2} b$
$(2) {(3 r - 2 k)}^{3} + {(3 r + 2 k)}^{3} = {(3 r)}^{3} - {(2 k)}^{3} - 3 \times {(3 r)}^{2} \times 2 k + 3 \times 3 r \times {(2 k)}^{2} + {(3 r)}^{3} + {(2 k)}^{3} + 3 \times {(3 r)}^{2} \times 2 k + 3 \times 3 r \times {(2 k)}^{2} = 27 r^{3} - 8 k^{3} - 54 r^{2} k + 36 r k^{2} + 27 r^{3} + 8 k^{3} + 54 r^{2} k + 36 r k^{2} = 54 r^{3} + 72 r k^{2}$
$(3) {(4 a - 3)}^{3} - {(4 a + 3)}^{3} = {(4 a)}^{3} - {(3)}^{3} - 3 \times {(49)}^{2} \times 3 + 3 \times 49 \times {(3)}^{2} - \{{(49)}^{3} + {(3)}^{3} + 3 \times {(49)}^{2} \times 3 + 3 \times 49 \times {(3)}^{2}\} = 64 a^{3} - 27 - 144 a^{2} + 108 - 64 a^{3} - 27 - 144 a^{2} - 108 = - 54 - 288 a^{2}$
$(4) {(5 x - 7 y)}^{3} + {(5 x + 7 y)}^{3} = {(5 x)}^{3} - {(7 y)}^{3} - 3 \times {(5 x)}^{2} \times 7 y + 3 \times 5 x \times {(7 y)}^{2} + {(5 x)}^{3} + {(7 y)}^{3} + 3 \times {(5 x)}^{2} \times (7 y) + 3 \times 5 x \times {(7 y)}^{2} = 125 x^{3} - 343 y^{3} - 525 x^{2} y + 735 x y^{2} + 125 x^{3} + 343 y^{3} + 525 x^{2} y + 735 x y^{2} = 250 x^{3} + 1470 x y^{2}$

Page No 28:

Question 1:

Expand.
(1) (2p + q +5)²
(2) (m + 2n +3r)²
(3) (3x +4y − 5p)²
(4) (7m − 3n − 4k)²

Answer:

It is known that,
${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
$(1) {(2 p + q + 5)}^{2} = {(2 p)}^{2} + {(q)}^{2} + {(5)}^{2} + 2 \times 2 p \times q + 2 \times q \times 5 + 2 \times 5 \times 2 p = 4 p^{2} + q^{2} + 25 + 4 p q + 10 q + 20 p$
$(2) {(m + 2 n + 3 r)}^{2} = {(m)}^{2} + {(2 n)}^{2} + {(3 r)}^{2} + 2 \times m \times 2 n + 2 \times 2 n \times 3 r + 2 \times 3 r \times m = m^{2} + 4 n^{2} + 9 r^{2} + 4 m n + 12 n r + 6 m r$
$(3) {(3 x + 4 y - 5 p)}^{2} = {(3 x)}^{2} + {(4 y)}^{2} + {(- 5 p)}^{2} + 2 \times 3 x \times 4 y + 2 \times 4 y \times (- 5 p) + 2 \times (- 5 p) \times 3 x = 9 x^{2} + 16 y^{2} + 25 p^{2} + 24 x y - 40 y p - 30 p x$
$(4) {(7 m - 3 n - 4 k)}^{2} = {(7 m)}^{2} + (- 3 n^{2}) + {(- 4 k)}^{2} + 2 \times 7 m \times (- 3 n) + 2 \times (- 3 n) \times (- 4 k) + 2 \times (- 4 k) \times 7 m = 49 m^{2} + 9 n^{2} + 16 k^{2} - 42 m n + 24 n k - 56 k m$

Page No 28:

Question 2:

Simplify.
(1) (x − 2y +3)² + (x + 2y − 3)²
(2) (3k − 4r − 2m)²− (3k + 4r − 2m)²
(3) (7a − 6b + 5c)²+ (7a + 6b − 5c)²

Answer:

It is known that,
${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
$(1) {(x - 2 y + 3)}^{2} + {(x + 2 y - 3)}^{2} = {(x)}^{2} + {(- 2 y)}^{2} + {(3)}^{2} + 2 \times x \times (- 2 y) + 2 \times (- 2 y) \times 3 + 2 \times 3 \times x + {(x)}^{2} + {(2 y)}^{2} + {(- 3)}^{2} + 2 \times x \times 2 y + 2 \times (2 y) \times (- 3) + 2 \times (- 3) \times x = x^{2} + 4 y^{2} + 9 - 4 x y - 12 y + 6 x + x^{2} + 4 y^{2} + 9 + 4 x y - 12 y - 6 x = 2 x^{2} + 8 y^{2} + 18 - 24 y$
$(2) {(3 k - 4 r - 2 m)}^{2} - {(3 k + 4 r - 2 m)}^{2} = {(3 k)}^{2} + {(- 4 r)}^{2} + {(- 2 m)}^{2} + 2 \times 3 k \times (- 4 r) + 2 \times (- 4 r) \times (- 2 m) + 2 \times (- 2 m) \times 3 k - \{{(3 k)}^{2} + {(4 r)}^{2} + {(- 2 m)}^{2} + 2 \times 3 k \times 4 r + 2 \times 4 r \times (- 2 m) + 2 \times (- 2 m) \times 3 k\} = 9 k^{2} + 16 r^{2} + 4 m^{2} - 24 k r + 16 r m - 12 m k - 9 k^{2} - 16 r^{2} - 4 m^{2} - 24 k r + 16 r m + 12 m k = 32 r m - 48 k r$
$(3) {(7 a - 6 b + 5 c)}^{2} + {(7 a + 6 b - 5 c)}^{2} = {(7 a)}^{2} + {(- 6 b)}^{2} + {(5 c)}^{2} + 2 \times 7 a \times (- 6 b) + 2 \times (- 6 b) \times 5 c + 2 \times 5 c \times 7 a + {(7 a)}^{2} + {(6 b)}^{2} + {(- 5 c)}^{2} + 2 \times 7 a \times 6 b + 2 \times 6 b \times (- 5 c) + 2 \times (- 5 c) \times 7 a = 49 a^{2} + 36 b^{2} + 25 c^{2} - 84 a b - 60 b c + 70 c a + 49 a^{2} + 36 b^{2} + 25 c^{2} + 84 a b - 60 b c - 70 c a = 98 a^{2} + 72 b^{2} + 50 c^{2} - 120 b c$

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