Question

Find all the integral values of $a$ for which the quadratic equation $(x-a)(x-10)+1=0$ has integral roots.

• A. $12,8$
• B. $11,9$
• C. $12,9$
• D. $11,8$

Answer 1

The correct option is A $12,8$

Here the equation is $x^2-(a+10)x+10a+1=0.$ Since integral roots will always be rational it means $D$ should be a perfect square.
From (i) $D=a^2-20a+96.$
$\Rightarrow$ $D={(a-10)}^2-4$ $\Rightarrow$ $4={(a-10)}^2-D$
If $D$ is a perfect square it means we want difference of two perfect square as $4$ which is possible only when ${(a-10)}^2=4$ and $D=0.$
$\Rightarrow$ $(a-10)=\pm 2$ $\Rightarrow$ $a=12,8$