Factoring...?

Algebra Level 3

$n=x^2+10x+25$

In the equation above, $x$ is an integer if and only if $\text{\_\_\_\_\_\_\_\_\_\_}$ .

n

is an even number

n

is an odd number

\sqrt{n}

is an integer

n^2

is an integer

n

is an integer

2 solutions

James Dohm
Jul 18, 2016

Observe that $n=(x+5)^2$ $\rightarrow$ $\pm \sqrt{n}-5=x$ .

1) $x = -5 \pm \sqrt{n}$
2) You should add that, n > 0

A Former Brilliant Member - 4 years, 11 months ago

Thanks, added (1). (2) does not need to be mentioned for this problem.

James Dohm - 4 years, 11 months ago

Majed Kalaoun
Jul 4, 2017

1. $n=x^2+10x+25\Rightarrow n=(x+5)^2$

2. $\Rightarrow \pm\sqrt{n}-5=x$

Therefore, from the above, we can conclude that $x$ is an integer if, and only if $\sqrt{n}$ is an integer.