Question

For what values of $a$, do the roots $x_1$ and $x_2$ of the equation $x^2-(3a+2)x+a^2=0$ satisfy the relation $x_1=9x_2$? Find the roots.

Answer 1

Given that the roots of the given equation is $x_1$ and $x_2$

Sum of the roots is $\frac{3a+2}{1}$

$\Rightarrow x_1+x_2=3a+2$

$\Rightarrow 9x_2+x_2=3a+2$ ..... $(x_1=9x_2)$

$\Rightarrow 10x_2=3a+2$

$\Rightarrow x_2=\frac{3a+2}{10}$ ......... $\left(1\right)$

Product of the root is $\frac{a^2}{1}$

$x_1\times x_2=a^2$

$9x_2^2=a^2$

$x_2^2=\frac{a^2}{9}$

$x_2=\frac{a}{3}$ ........ $\left(2\right)$

Put the value of $\left(2\right)$ in $\left(1\right)$, we get

$\frac{3a+2}{10}$ $=\frac{a}{3}$

$9a+6=10a$

$\Rightarrow a=6$

$\Rightarrow x_2=2$ and $x_1=18$