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

We first calculate the expression $(x + 5)^2 - (x - 5)^2$.

Using the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$, let:

• $a = (x + 5)$
• $b = (x - 5)$

Now we evaluate:

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

This simplifies as follows:

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

Putting it all together, we have: $10(2x) = 20x$

We can clearly see that $20x$ is divisible by 4, since: $20x = 4(5x)$

For part (b)(i):

To factorise $25x^2 - 49n^2$, we again use the difference of squares:

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

Here, we can identify:

• $a^2 = (5x)^2$
• $b^2 = (7n)^2$

Thus, we factor as follows:

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

For part (b)(ii):

To factorise $2x^2 - 9x - 18$, we look for two numbers that multiply to $2(-18) = -36$ and add up to $-9$. These numbers are $-12$ and $3$.

We rewrite the middle term: $2x^2 - 12x + 3x - 18$

Now, we can group the terms: $= 2x(x - 6) + 3(x - 6)$

Factoring out the common factor $(x - 6)$: $= (2x + 3)(x - 6).$