Answer
x = 3
or
x= -1
Both the values satisfy the distance condition.
Work Step by Step
To find x, we will calculate distance between given points, i. e. (x, 2) and (1, 5), using distance formula and equate it to $\sqrt 13$. Thus-
$ \sqrt { (x_{2} - x_{1}) ^{2} + (y_{2} - y_{1}) ^{2}}$ = $\sqrt 13$
$ \sqrt { (1 - x) ^{2} + (5- 2) ^{2}}$ = $\sqrt 13$
$ \sqrt { (1 -2x + x^{2}) + 9}$ = $\sqrt 13$
Squaring on both sides, we get-
$ { (1 -2x + x^{2}) + 9}$ = $13$
$ x^{2}-2x + 10$ = $13$
OR
$ x^{2}-2x-3$ = 0 ( adding -13 on both sides)
$(x-3) (x+1)$ = 0 (On factorizing)
Equating each factor to zero, we get-
$x= 3 or -1$
Therefore x may be 3 or -1
