Question

The roots of the equation \[{{x}^{2}}-2px+{{p}^{2}}+{{q}^{2}}+2qr+{{r}^{2}}=0\left( p,q,r\in \mathbb{Z} \right)\] are
(a) Rational and different
(b) Rational and equal
(c) Irrational
(d) Imaginary

Answer 1

Hint: In this type of question we have to use the concept of quadratic equation. We know that the general form of the quadratic equation is given by, \[a{{x}^{2}}+bx+c=0\]. Also we know that, the roots of the quadratic equation is given by, \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] where \[D={{b}^{2}}-4ac\] is called the discriminant of the equation. We can comment on the nature of the roots from the obtained discriminant value.

Complete step by step answer:
Now, we have to obtain the nature of the roots of the equation \[{{x}^{2}}-2px+{{p}^{2}}+{{q}^{2}}+2qr+{{r}^{2}}=0\left( p,q,r\in \mathbb{Z} \right)\]
We know that the roots of the quadratic equation \[a{{x}^{2}}+bx+c=0\] are given by \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] where \[D={{b}^{2}}-4ac\] is called the discriminant and depending on the value of \[D\] the nature of the roots can be identified as follows:
If \[D<0\] then the quadratic equation has complex roots.
If \[D>0\] then the quadratic equation has real and distinct roots.
If \[D=0\] then the quadratic equation has real and equal roots.
Now, the given quadratic equation is \[{{x}^{2}}-2px+{{p}^{2}}+{{q}^{2}}+2qr+{{r}^{2}}\] comparing it with the general form \[a{{x}^{2}}+bx+c=0\] we can write,
\[\Rightarrow a=1,b=-2p,c={{p}^{2}}+{{q}^{2}}+2qr+{{r}^{2}}\]
\[\begin{align}
  & \Rightarrow D={{b}^{2}}-4ac \\
 & \Rightarrow D={{\left( -2p \right)}^{2}}-4\times 1\times \left( {{p}^{2}}+{{q}^{2}}+2qr+{{r}^{2}} \right) \\
 & \Rightarrow D=4{{p}^{2}}-4{{p}^{2}}-4{{q}^{2}}-8qr-4{{r}^{2}} \\
 & \Rightarrow D=-4\left( {{q}^{2}}+2qr+{{r}^{2}} \right) \\
 & \Rightarrow D=-4{{\left( q+r \right)}^{2}} \\
 & \Rightarrow D<0 \\
\end{align}\]
Since, \[D<0\] the given equation has complex roots.

So, the correct answer is “Option d”.

Note: In this type of question students have to remember the conditions for the nature of the roots of the quadratic equation using the value of the discriminant. Also students can solve this question by finding the roots of the equation using the direct formula which also gives the same answer.