The value of 'a' for which one root of the quadratic equation [ ( a^2 - 5a + 3 )x^2 + ( 3a…

Question

The value of '$a$' for which one root of the quadratic equation
$$ \left( {{a^2} - 5a + 3} \right){x^2} + \left( {3a - 1} \right)x + 2 = 0$$

Accepted Answer

Let the roots of given equation be $$\alpha $$ and $2$ $\alpha $then$\alpha  + 2\alpha  = 3\alpha  = {{1 - 3a} \over {{a^2} - 5a + 3}}$and$\alpha .2\alpha  = 2{\alpha ^2} = {2 \over {{a^2} - 5a + 3}}$$ \Rightarrow \alpha  = {{1 - 3a} \over {3\left( {{a^2} - 5a + 3} \right)}}$$\therefore$$2\left[ {{1 \over 9}{{{{\left( {1 - 3a} \right)}^2}} \over {{{\left( {{a^2} - 5a + 3} \right)}^2}}}} \right]$$ = {2 \over {{a^2} - 5a + 3}}$${{{{\left( {1 - 3a} \right)}^2}} \over {\left( {{a^2} - 5a + 3} \right)}} = 9$or$9{a^2} - 6a + 1$$ = 9{a^2} - 45a + 27$or$39a = 26$or$a = {2 \over 3}$

Solutions · 1