Question

After solving the equation $\frac{(5x+2{)}^2+(5x-2{)}^2}{(5x+2{)}^2-(5x-2{)}^2}=\frac{13}{12}$, solution set for $x$ is $\{m,0.27\}$. Then, the value of $m$ is:

Answer 1

Given $\frac{{(5x+2)}^2+{(5x-2)}^2}{{(5x+2)}^2-{(5x-2)}^2}=\frac{13}{12}$
Applying the formulae, ${(a+b)}^2+{(a-b)}^2=2{(a}^2+b^2)$
and ${(a+b)}^2-{(a-b)}^2=4ab$ where $a=5x;b=2$
$\frac{{2\left(\right(5x)}^2+2^2)}{4\times \left(5x\right)\times 2}=\frac{13}{12}$
$\frac{{25x}^2+4}{20x}=\frac{13}{12}$
On cross multiplying and simplifying,
$75x^2-65x+12=0$
$75x^2-20x-45x+12=0$
$5x(15x-4)-3(15x-4)=0$
$(5x-3)(15x-4)=$
$x=\left\{\frac{3}{5},\frac{4}{15}\right\}=\{0.6,0.27\}$
Hence, $m=0.6$