Question

For any integer x, x^2-x will always produce an even value. True or false

Answer 1

That true. You have two ways to show it:

Method 1:

Factor the expression. You have

$x^2-x = x(x-1)$

So, you're multiplying two consecutive numbers. One of them must be even, and if a factor of a product is even, the result will be even.

Method 2:

If x is even, $x^2$ is even as well. So, $x^2-x$ is a subtraction between even numbers, which gives an even number.

If x is odd, $x^2$ is odd as well. So, $x^2-x$ is a subtraction between odd numbers, which gives an even number.