First slide

Theory of equations

Question

Suppose a $a \in I$ and the equation $(x - a) (x - 5) = 3$ has integral roots, then the set of values which a can take is:

Moderate

A

$ϕ$
B

$\{- 11, - 13\}$
C

$\{3, 7\}$
D

$\{- 3, - 7\}$

Solution

Let $m \in I$ be a roots of $(x - a) (x - 5) = 3$ , then
$(m - a) (m - 5) = 3$
As $m - a$ and $m - 5$ are integers, $m - 5 = \pm 1$ or $m - 5 = \pm 3$
$\Rightarrow m = 2, 4, 6, 8$
Thus, $a = 3, 7$

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