Question

Simplify (7a /2-5b/2)^2-(5a/2-7b/2)^2 and find the value for a =1 and b=-2)

Answer 1

Answer:

$\left( \frac{7a}{2} - \frac{5b}{2}\right)^{2} - \left( \frac{5a}{2} - \frac{7b}{2}\right)^{2}$

$=[ \left( \frac{7a}{2} - \frac{5b}{2}\right)+\left( \frac{5a}{2} - \frac{7b}{2}\right)][ \left( \frac{7a}{2} - \frac{5b}{2}\right)-\left( \frac{5a}{2} - \frac{7b}{2}\right)]$

$\boxed { \pink { a^{2} - b^{2} = (a+b)(a-b) }}$

$= \left( \frac{7a}{2} - \frac{5b}{2}+\frac{5a}{2} - \frac{7b}{2}\right)\left( \frac{7a}{2} - \frac{5b}{2}-\frac{5a}{2} + \frac{7b}{2}\right)$

$= \left( \frac{7a}{2} + \frac{5a}{2} - \frac{5b}{2} - \frac{7b}{2} \right) \left( \frac{7a}{2} - \frac{5a}{2} - \frac{5b}{2} +\frac{7b}{2} \right)$

$= \left( \frac{7a+5a}{2} - \frac{5b+7b}{2} \right) \left( \frac{7a-5a}{2} + \frac{-5b+7b}{2}\right)$

$= \left( \frac{ 12a}{2} - \frac{12b}{2}\right) \left( \frac{2a}{2} + \frac{2b}{2} \right)$

$= ( 6a-6b)(a+b)$

$= 6(a-b)(a+b)$

$= 6(a^{2} - b^{2})$

$Now, \: If \: a = 1 \:and \: b = -2 \: then \\Value \: of \: \left( \frac{7a}{2} - \frac{5b}{2}\right)^{2} - \left( \frac{5a}{2} - \frac{7b}{2}\right)^{2}\\= 6(a^{2} - b^{2}) \\= 6[ 1^{2} - (-2)^{2} ] \\= 6( 1- 4) \\= 6 (-3) = -18$

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