Question

Consider the quadratic equation $n{x}^2+7\sqrt{n}x+n=0$, where $n$ is a positive integer. Which of the following statements are necessarily correct?
I. For any $n$, the roots are distinct.
II. There are infinitely many values of $n$ for which both roots are real.
III. The product of the roots is necessarily an integer.

• A. III only
• B. I and III only
• C. II and III only
• D. I, II and III

Answer 1

Answer: (B)

I and III only

Let the discriminant of the quadratic equation $a{x}^2+bx+c$, where $a,b$ and $c$ are real, be given by $D=b^2-4ac$. Roots of the given equation are

i) real and distinct if $D>0$

ii) real and equal if $D=0$

iii) imaginary and distinct if $D<0$

Product of the roots is $\frac{c}{a}$.

For the given problem, $D=49n-4n^2$

I) roots are distinct if $D\ne 0$

$\Rightarrow 49n\ne 4n^2$

$\Rightarrow n\ne \frac{49}{4}$

$n$ is a positive integer, so for all n, the roots are distinct.

II) Both roots are real $\Rightarrow D\ge 0$

$\Rightarrow 49n\ge 4n^2$

$\Rightarrow n\le \frac{49}{4}$

So, such values of $n$ are not infinite.

III) Product of roots $=\frac{n}{n}=1$, which is an integer.