Answer:
(i) \[{{(3x+7)}^{2}}-84x={{(3x-7)}^{2}}\] L.H.S. \[={{(3x+7)}^{2}}-84x\] \[=\{{{(3x)}^{2}}+2(3x)\,(7)+{{(7)}^{2}}\}\,-84x\] \[=(9{{x}^{2}}+42x+49)\,-84x\] \[=9{{x}^{2}}+(42x-84x)+49\] |Combining the like terms \[=9{{x}^{2}}-42x+49\] ?(1) R.H.S. \[={{(3x-7)}^{2}}\] \[={{(3x)}^{2}}-2(3x)\,(7)\,+{{(7)}^{2}}\] \[=9{{x}^{2}}-42x+49\] ?(2) From equations (1) and (2), \[{{(3x+7)}^{2}}-84x={{(3x-7)}^{2}}\] (ii) \[{{(9p-5q)}^{2}}+180pq={{(9p+5q)}^{2}}\] L.H.S.\[={{(9p-5q)}^{2}}+180pq\] \[=\{{{(9p)}^{2}}-2(9p)\,(5q)+{{(5q)}^{2}}\}\,+180pq\] \[=(81{{p}^{2}}-90pq+25{{q}^{2}})\,+180pq\] \[=81{{p}^{2}}+(180pq-90pq)+25{{q}^{2}}\] |Combing the like terms \[=81{{p}^{2}}+90pq+25{{q}^{2}}\] ?(1) R.H.S. \[={{(9p+5q)}^{2}}\] \[={{(9p)}^{2}}+2(9p)(5q)\,+{{(5q)}^{2}}\] \[=81{{p}^{2}}+9pq+25{{q}^{2}}\] ?(2) From equations. (1) and (2) \[{{(9p-5q)}^{2}}+180pq={{(9p+5q)}^{2}}\] (iii) \[{{\left( \frac{4}{3}m-\frac{3}{4}n \right)}^{2}}+2mn=\frac{16}{9}\,{{m}^{2}}+\frac{9}{16}{{n}^{2}}\] L.H.S. \[={{\left( \frac{4}{3}m-\frac{3}{4}n \right)}^{2}}+2mn\] \[={{\left( \frac{4}{3}m \right)}^{2}}\,-2\left( \frac{4}{3}m \right)\left( \frac{3}{4}m \right)+{{\left( \frac{3}{4}m \right)}^{2}}+2mn\] \[=\frac{16}{9}{{m}^{2}}-2mn+\frac{9}{16}\,{{n}^{2}}+2mn\] \[=\frac{16}{9}\,{{m}^{2}}+(2mn-2mn)\,+\frac{9}{16}\,{{n}^{2}}\] |Combining the like terms \[=\frac{16}{9}{{m}^{2}}+\frac{9}{16}\,{{n}^{2}}\] = R.H.S. (iv) \[(4pq)+3q{{)}^{2}}-{{(4pq-3q)}^{2}}=48p{{q}^{2}}\] L.H.S. \[={{(4pq+3q)}^{2}}-{{(4pq-3q)}^{2}}\] \[=\{{{(4pq)}^{2}}+2(4pq)\,(3q)+{{(3q)}^{2}}\}\]\[-\{{{(4pq)}^{2}}-2(4pq)\,(3q)\,+{{(3q)}^{2}}\}\] \[=(16{{p}^{2}}{{q}^{2}}+24p{{q}^{2}}+9{{q}^{2}})\]\[-(16{{p}^{2}}{{q}^{2}}-24p{{q}^{2}}+9{{q}^{2}})\] \[=16{{p}^{2}}{{q}^{2}}+24p{{q}^{2}}+9{{q}^{2}}-16{{p}^{2}}{{q}^{2}}\]\[+24p{{q}^{2}}-9{{q}^{2}}\] \[=(16{{p}^{2}}{{q}^{2}}-16{{p}^{2}}{{q}^{2}})+(24p{{q}^{2}}+24p{{q}^{2}})\]\[+(9{{q}^{2}}-9{{q}^{2}})\] |Combining the like terms \[=48p{{q}^{2}}\] = R.H.S. (v) \[+(c-a)\,(c+a)=0\] L.H.S. \[=(a-b)\,(a+b)\,+(b-c)\,(b+c)\]\[+(c-a)\,(c+a)\] \[={{a}^{2}}-{{b}^{2}}+{{b}^{2}}-{{c}^{2}}+{{c}^{2}}-{{a}^{2}}\] |Using identity III \[=({{a}^{2}}-{{a}^{2}})+({{b}^{2}}-{{b}^{2}})+({{c}^{2}}-{{c}^{2}})\] |Combining the like terms = 0 = R.H.S.
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