Question

Consider the equation ${x^2} - 2x - n = 0$, where $n \in N$and $n \in \left[ {5,100} \right]$. The total number of different values of n so that the given equation has integral root is
A.8
B.3
C.6
D.4

Answer 1

Hint: We know that roots of the quadratic equation is $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ .In ${x^2} - 2x - n = 0$ b=-2 a=1 and c=-n. For roots to be integers then $D = {b^2} - 4ac$ must be positive and greater than 0. We will impose this condition and find the possible values of n which satisfy the given condition that is $n \in N$ and $n \in \left[ {5,100} \right]$.

Complete step-by-step answer:
given equation is ${x^2} - 2x - n = 0$where a =1 b=-2 and c=-n
roots of the equation= $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
$ = \dfrac{{2 \pm \sqrt {{2^2} - 4 \times 1 \times - n} }}{{2 \times 1}} = 1 \pm \sqrt {1 + n} $
For roots to be integers $1 + n$ must be a perfect square, positive and greater than or equal to 0
$\Rightarrow$ $1 + n \geqslant 0$
$\Rightarrow$ $n \geqslant - 1$
$\Rightarrow$ $n \in \left[ {5,100} \right]$
$\Rightarrow 1 + n \in \left[ {6,101} \right]$
$1 + n$ will be perfect square when
$1 + n$=9,16,25,36,49,64,81,100 then the respective values of n are
n=8,15,24,35,48,63,80,99
Answer is option (A)

Note: If $a{x^2} + bx + c = 0$
For real and positive roots $D = {b^2} - 4ac$ must be greater than zero.
For real and equal roots $D = {b^2} - 4ac$ must be zero.
For imaginary roots $D = {b^2} - 4ac$ must be less than zero.
Roots of the equation are $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$