(a) (i) Multiply out and simplify $(x + 5)^2$ - Junior Cycle Mathematics - Question 11 - 2016

We will evaluate the expression $(x + 5)^2 - (x - 5)^2$.

Using the difference of squares formula:

$a^2 - b^2 = (a - b)(a + b)$

Letting $a = (x + 5)$ and $b = (x - 5)$, we have:

$(x + 5) - (x - 5) = 10$ $(x + 5) + (x - 5) = 2x$

Thus, we can rewrite the expression as:

$(10)(2x) = 20x$

Since $20x$ can be factored as $4(5x)$, it is always divisible by 4.

For (i) $25x^2 - 49n^2$, we can apply the difference of squares:

$25x^2 - 49n^2 = (5x - 7n)(5x + 7n)$

For (ii) $2x^2 - 9x - 18$, we need to find two numbers that multiply to $2(-18) = -36$ and add to $-9$. The numbers $-12$ and $3$ work. We can therefore factor as:

$2x^2 - 12x + 3x - 18 = 2x(x - 6) + 3(x - 6) = (2x + 3)(x - 6)$