Question

If $\alpha$ and $\beta$ are the roots of the equation $4x^2-5x+2=0$, find the equation whose roots are

$\alpha +3\beta$ and $3\alpha +\beta$.

• A. $16x^2+80x+107=0$
• B. $16x^2-80x+107=0$
• C. $16x^2-80x-107=0$
• D. $16x^2+80x-107=0$

Answer 1

Answer: (B)

$16x^2-80x+107=0$

$4x^2-5x+2=0$

If $\alpha$ and $\beta$ are the roots of this equation,

then , sum of roots: $\alpha +\beta$ = $\frac{5}{4}$

Product of roots: $\alpha .\beta =\frac{2}{4}$
The equation which has roots as : $\alpha +3\beta$ and $\beta +3\alpha$

Sum of roots: $4\alpha +4\beta$ = $4\left(\frac{5}{4}\right)=5$

Product of roots: $(\alpha +3\beta )(3\alpha +\beta )$

$=3({\alpha }^2+{\beta }^2)+10\alpha \beta$

$=3(\alpha +\beta {)}^2-6\alpha \beta +10\alpha \beta$

$=3{\left(\frac{5}{4}\right)}^2+4\frac{2}{4}$

$=\frac{107}{16}$
Thus new equation is :$x^2-Sx+P=0$

$\therefore x^2-5x+\frac{107}{16}=0$

$\therefore 16x^2-80x+107=0$