Question

A student while solving a quadratic equation in $x$, he copied its constant term incorrectly and got its roots as $5$ and $9.$ Another student copied the constant term and coefficient of $x^2$ of the same equation correctly as $12$ and $4$ respectively. If $s,p$ and $\Delta$ denote respectively the sum of the roots, the product of the roots and the discriminate of the correct equation, then $\frac{\Delta }{3p+s}=$

• A. 48
• B. 45
• C. 8
• D. None of the above

Answer 1

Answer: (D)

Let the equation be $a{x}^2+bx+c=0$
Sum of roots $=-\frac{b}{a}$
and product of roots $=\frac{c}{a}$
when constant term is copied wrongly we get roots as 5 and 9 .
$\therefore -\frac{b}{a}=5+9$ [sum will remain correct $]$
$\Rightarrow -\frac{b}{a}=14$
Another student copied constant term and coefficient of
$x^2$ correctly us 12 and 4 .
$\therefore c=12$ and $a=4$
$\therefore \frac{-b}{a}=14\Rightarrow b=-56$
$[\because a=4]$
$\therefore S=-\frac{b}{a}=\frac{-(-56)}{4}=14$
$P=\frac{c}{a}=\frac{12}{4}=3$
$\Delta =b^2-4ac$
$=(-56{)}^2-4\times 4\times 12$
$=3136-192=2944$
$\therefore \frac{\Delta }{3p+S}=\frac{2944}{3\times 3+14}=\frac{2944}{23}=128$