Question:
Find the value
$x^{2}+\frac{12}{35} x+\frac{1}{35}$
Solution:
Splitting the middle term,
$=x^{2}+\frac{5}{35} x+\frac{7}{35} x+\frac{1}{35}$
$\left[\therefore \frac{12}{35}=\frac{5}{35}+\frac{7}{35}\right.$ and $\left.\frac{5}{35} \times \frac{7}{35}=\frac{1}{35}\right]$
$=x^{2}+x / 7+x / 5+1 / 35$
$=x(x+1 / 7)+1 / 5(x+1 / 7)$
$=(x+1 / 7)(x+1 / 5)$
$\therefore \mathrm{x}^{2}+\frac{12}{35} \mathrm{x}+\frac{1}{35}=\left(\mathrm{x}+\frac{1}{7}\right)\left(\mathrm{x}+\frac{1}{5}\right)$