Show that: |
(a) \[{{(3x+7)}^{2}}-84x={{(3x-7)}^{2}}\] |
(b) \[{{\left( 9p-5q \right)}^{2}}+180\text{ }pq={{\left( 9p+5q \right)}^{2}}\] |
(c) \[{{\left( \frac{4}{3}m-\frac{3}{4}n \right)}^{2}}+2mn=\frac{16}{9}{{m}^{2}}+\frac{9}{16}{{n}^{2}}\] |
(d) \[\,{{(4pq+3q)}^{2}}-{{(4pq-3q)}^{2}}=48p{{q}^{2}}\] |
Answer:
(a) \[{{\left( 3x+7 \right)}^{2}}84x={{\left( 3x7 \right)}^{2}}\] Taking L.H.S. \[={{\left( 3x+7 \right)}^{2}}84x\] \[=\left\{ {{\left( 3x \right)}^{2}}+2\left( 3x \right)\times 7+{{\left( 7 \right)}^{2}} \right\}84x\] \[=\text{ }9{{x}^{2}}+42x+4984x\] \[=9{{x}^{2}}+4942x\] \[={{\left( 3x \right)}^{2}}\left( 2\times 3\times 7 \right)x+{{\left( 7 \right)}^{2}}\] \[=9{{x}^{2}}42x+49\] \[={{\left( 3x7 \right)}^{2}}=\text{ }R.H.S\] (b) \[{{\left( 9p5q \right)}^{2}}+180pq={{\left( 9p+5q \right)}^{2}}\] Taking L.H.S. \[={{\left( 9p5q \right)}^{2}}+180pq\] \[=\{{{\left( 9p \right)}^{2}}2\times 9p\times 5q+{{\left( 5q \right)}^{2}}+180\text{ }pq\] \[=81{{p}^{2}}90pq+25{{q}^{2}}+180pq\] \[=\text{ }81{{p}^{2}}+25{{q}^{2}}+90pq\] \[={{\left( 9p \right)}^{2}}+{{\left( 5q \right)}^{2}}+\left( 2\times 9\times 5 \right)pq\] \[=\text{ }9{{p}^{2}}+25{{q}^{2}}+90pq\] \[={{\left( 9p+5q \right)}^{2\text{ }}}=\text{ }R.H.S\] (c) \[{{\left( \frac{4}{3}m-\frac{3}{4}n \right)}^{2}}+2mn=\frac{16}{9}{{m}^{2}}+\frac{9}{16}{{n}^{2}}\] Taking L.H.S. \[={{\left( \frac{4}{3}m-\frac{3}{4}n \right)}^{2}}+2mn\] \[=\left\{ \left( \frac{4}{3}{{m}^{2}} \right)-2\left( \frac{4}{3}m \right)\left( \frac{3}{4}n \right)+{{\left( \frac{3}{4}n \right)}^{2}} \right\}+2mn\] \[=\frac{16}{9}{{m}^{2}}-2mn+\frac{9}{16}{{n}^{2}}+2mn\] \[=\frac{16}{9}{{m}^{2}}+\frac{9}{16}{{n}^{2}}-2mn+2mn\] \[=\frac{16}{9}{{m}^{2}}+\frac{9}{16}{{n}^{2}}=\,\]R.H.S. (d) \[{{(4pq+3q)}^{2}}-{{(4pq-3q)}^{2}}=48p{{q}^{2}}\] Taking L.H.S \[={{\left( 4pq+3q \right)}^{2}}{{\left( 4pq3q \right)}^{2}}\] \[=\{{{(4pq)}^{2}}+2(4pq)(3q)+{{(3q)}^{2}}\] \[~-\left\{ {{\left( 4pq \right)}^{2}}2\left( 4pq \right)3q+{{\left( 3q \right)}^{2}} \right\}\] \[=16{{p}^{2}}{{q}^{2}}+24p{{q}^{2}}+9{{q}^{2}}\left( 16{{p}^{2}}{{q}^{2}}24p{{q}^{2\text{ }}}+9{{q}^{2}} \right)\] \[=(16-16){{p}^{2}}{{q}^{2}}+(24+24)\text{ }p{{q}^{2}}+\text{(}9q)\text{ }{{q}^{2}}\] \[=0+48p{{q}^{2}}+0\] \[=\text{ }48\text{ }p{{q}^{2}}\text{ }=\]R.H.S Total number of shoes = 400
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