Question

Number of different values of p for which the equation $x^2-2(p+q)x+(q^2+2qp)=0$ has integral roots is $(p,qϵN,pϵ[5,10])$

• A. 5
• B. 6
• C. 0
• D. 4

Answer 1

The correct option is B

6

The roots of the equation are $\frac{2(p+q)\mp \sqrt{\left(2\right(p+q){)}^2-4(q^2+2pq)}}{2}$

$=\frac{2(p+q)\mp \sqrt{4(p+q{)}^2-4(q^2+2pq)}}{2}=\frac{2(p+q)\mp 2\sqrt{p^2+q^2+2pq-q^2-2pq}}{2}=\frac{2(p+q)\mp 2p}{2}=p+q\mp p$

$\Rightarrow$ The roots will be an integer for all values of p $ϵ$ [5, 10] and p $ϵ$ N, which is 6