Question

If $\alpha ,\beta$ are the roots of equation $x^2-px+q=0,$ then find the equation the roots of which are $\left({\alpha }^2{\beta }^2\right)and\alpha +\beta$.

• A. $x^2-(p+q^2)x+p{q}^2$
• B. $x^2-(p+q)x+p{q}^2$
• C. $x^2-(p+q^2)x+pq$
• D. $x^2-(p^2+q^2)x+p^2{q}^2$

Answer 1

The correct option is A $x^2-(p+q^2)x+p{q}^2$

Given : $\alpha ,\beta$ are the roots of equation $x^2-px+q=0$

sum of roots = $\alpha +\beta =\frac{-(-p)}{1}=p$ (If $a{x}^2+bx+c=0$ has $\alpha$ and $\beta$ are roots then sum of roots $=(-b)/a=\alpha +\beta$ product of roots = $c/a$)

product of roots = $\alpha \beta =\frac{q}{1}=q$

${\alpha }^2{\beta }^2=q^2$

To find : The equation whose roots are ${\alpha }^2{\beta }^2$ & $\alpha +\beta$

Equation : $(x-{\alpha }^2{\beta }^2)(x-(\alpha +\beta \left)\right)=0$

$\Rightarrow (x-q^2)(x-p)=0$

$x^2-px-q^{2}x+q^{2}p=0$

$x^2-(p+q^2)x+q^{2}p=0$

$\therefore$ equation is $x^2-(p+q^2)x+q^{2}p=0$