Question

If $m,n$ are the roots of the quadratic equation $x^2-3x+5=0,$ then the equation whose roots are $(m^2-3m+7)\&(n^2-3n+7)=$

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

Answer 1

The correct option is B $x^2-4x+4=0$

Given quadratic equation: $x^2-3x+5=0,$ and $m,n$ are it's roots.
$\Rightarrow m^2-3m+5=0\&n^2-3n+5=0$

$\therefore m^2-3m=-5\cdots \left(i\right)n^2-3n=-5\cdots \left(ii\right)$

Now, roots of the required equation are $(m^2-3m+7)\text{and}(n^2-3n+7)$
$\Rightarrow m^2-3m+7=-5+7=2\&n^2-3n+7=-5+7=2$
Hence, the roots of the required quadratic equation are $2,2$
$\because$ Sum of roots $=2+2=4$
and the product of roots $=2\times 2=4$
$\therefore$ The required equation is $x^2-4x+4=0$