Question

If $\alpha ,\beta$ are the roots of $x^2+3x+4=0,$ then the quadratic equation whose roots are ${\alpha }^3+4{\alpha }^2+7\alpha +7$ and ${\beta }^3-{\beta }^2-8\beta -14$ is

• A. $x^2+3x+4=0$
• B. $x^2+5x+6=0$
• C. $x^2-5x+6=0$
• D. $x^2-3x+4=0$

Answer 1

The correct option is C $x^2-5x+6=0$

Since, $\alpha ,\beta$ are the roots of $x^2+3x+4=0$
$\therefore {\alpha }^2+3\alpha +4=0...\left(1\right)$
and ${\beta }^2+3\beta +4=0...\left(2\right)$

Dividing ${\alpha }^3+4{\alpha }^2+7\alpha +7$ by ${\alpha }^2+3\alpha +4$, we get
${\alpha }^3+4{\alpha }^2+7\alpha +7=({\alpha }^2+3\alpha +4)(\alpha +1)+3$
$=3\left[\text{From}\right(1\left)\right]$

Dividing ${\beta }^3-{\beta }^2-8\beta -14$ by ${\beta }^2+3\beta +4$, we get
${\beta }^3-{\beta }^2-8\beta -14=({\beta }^2+3\beta +4)(\beta -4)+2$
$=2\left[\text{From}\right(2\left)\right]$

Sum of roots of required equation $=3+2=5$
Product of roots of required equation $=3\times 2=6$
Therefore, required equation is $x^2-5x+6=0$