I'm working from a textbook and am unsure if it's wrong (its probably me);
Q) Find the lengths of the line segment joining this pair, assume a> 0.
$(a+1,2a+3)$ and $(a-1,2a-1)$
Following the equation;
$L =\sqrt {(x_2-x_1)^2+(y_2-y_1)^2}$
With;
$x_2 = a-1$, $ x_1 = a+1$, $ y_2=2a-1, y_1=2a+3$
I get;
$L= \sqrt {((a-1)-(a+1))^2+((2a-1)-(2a+3))^2}$
Which to me means;
$L= \sqrt {(0)^2(2)^2}$
So,
$L= 2$
However, the book says;
$L=\sqrt{20}$
Where have I gone wrong?
\begin{align} \sqrt{((a-1) - (a+1))^2 + ((2a-1) - (2a+3))^2} &= \sqrt{(a-1-a-1)^2 + (2a-1-2a-3)^2}\\[0.3cm] &= \sqrt{(-2)^2 + (-4)^2}\\[0.3cm] &= \sqrt{4 + 16}\\[0.3cm] &= \sqrt{20}\\[0.3cm] &= 2\sqrt{5} \end{align}
I can tell based on your question you'll probably need some more details explained. Your very first step is correct, when you said $$ L = \sqrt{((a-1) - (a+1))^2 + ((2a-1) - (2a+3))^2}. $$
But I have no idea how you got from there to $\sqrt{(0)^2(2)^2}$. Also, that would equal 0, not 2. Perhaps did you mean $\sqrt{(0)^2 + (2)^2}$? That's close but still not quite right.
In any case, let's start from the beginning and look at one piece at a time. The first piece we'll look at is the $((a-1) - (a+1))^2$. The very first thing we need to do is distribute the minus sign into the $(a+1)$. So we get: $$ ((a-1) - (a+1))^2 = ((a-1) -a-1)^2 $$
The parentheses around the $a-1$ are unnecessary, so we get: $$ ((a-1) -a-1)^2 = (a-1 -a-1)^2 $$
Combine like terms, and we get: $$ (a-1-a-1)^2 = (-2)^2 $$
Simplify, and we get: $$ (-2)^2 = 4 $$
Now we look at $((2a-1) - (2a+3))^2$. We'll do the exact same process. First we distribute the minus sign into the $(2a+3)$. So we get: $$ ((2a-1) - (2a+3))^2 = ((2a-1) - 2a - 3)^2 $$
Next we drop the unnecessary parentheses on the $2a-1$, and we get: $$ ((2a-1) - 2a - 3)^2 = (2a-1 - 2a-3)^2$$
Then we combine like terms to get: $$ (2a-1-2a-3)^2 = (-4)^2 $$
And simplify to get: $$ (-4)^2 = 16 $$
To summarize, we have $((a-1) - (a+1))^2 = 4$ and $((2a-1) -(2a+3))^2 = 16$. Therefore: $$ \sqrt{((a-1) - (a+1))^2 + ((2a-1) - (2a+3))^2} = \sqrt{4 + 16} = \sqrt{20} $$
And this can further be simplified to $2\sqrt{5}$ but based on your textbook's answer I guess you aren't expected to do that.