prove that:
${(3 x + 7)}^{2} - {(3 x - 7)}^{2} = 84 x$

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Solution

We know that, Binomial expansion is,
$(x + y)^{n} = n_{c_{0}} x^{n} + n_{c_{1}} x^{n - 1} y + n_{c_{2}} x^{n - 2} y^{2} + . . + n_{c_{n}} y^{n}$
Now lets solve the given expansion,
${(3 x + 7)}^{2} - {(3 x - 7)}^{2}$
$= (^{2} C_{0} {(3 x)}^{2} +^{2} C_{1} (3 x) (7) +^{2} C_{2} {(7)}^{2}) - (^{2} C_{0} {(3 x)}^{2} -^{2} C_{1} (3 x) (7) +^{2} C_{2} {(7)}^{2})$
$({(3 x)}^{2} + 2 (21 x) + 49) - (9 x^{2} - 42 x + 49)$
$9 x^{2} + 42 x + 49 - 9 x^{2} + 42 x - 49$ $= 84 x$
Alternate method:
Lets take LHS and then equate it to RHS.
LHS $= {(3 x + 7)}^{2} - {(3 x - 7)}^{2}$
$= (3 x + 7 + 3 x - 7) (3 x + 7 - 3 x - 7)$ [ $∵ (a^{2} - b^{2} = (a + b) (a - b)$ ]
$= (6 x) (14)$
$= 84 x$
$=$ RHS
Hence, ${(3 x + 7)}^{2} - {(3 x - 7)}^{2} = 84 x$ proved.

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