question_answer

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