How di I do this algebra?

[iggymiggy]

How can I reduce this:
(2a X Sqrt(2))^2 + 4a^2 = (2a + 2b)^2

To this:
2a(Sqrt(3) − 1) = 2b

-Thanks

[factor-fiction]

Do it in steps. i'll start you off.
The left hand side becomes 8a^2 +4a^2=12a^2
Expand the right hand side: 4a^2 +4b^2 +8ab

Now see where that gets you.
I haven't done any more

[factor-fiction]

Okay, I've finished this but i'll let you have ago first. You need to rearrange into a quadratic equation =0
You can then solve it - i used the quadratic formula.

[iggymiggy]

I've had a go, and got no idea why I'd need to solve via quadratic.

Please help!

[MarkRae]

When is your homework due?

[iggymiggy]

It's not really homework, but it's part of a solution for working out the minimum cation-anion radius ratio for a coordination number of 8. It's just that I can't see how to get from the first line to the second line.

Many thanks all.

[factor-fiction]

You should have:

8a² = 4b² + 8ab
Rewrite as:
8a²-8ab-4b² = 0
Divide everything by 8
a ² -ab –(½)b² = 0


Now use the quadratic formula to solve it for a. The answer will be in terms of b
Then rearrange to get b

[factor-fiction]

I should have made it clearer that
a ² -ab –(½)b² = 0
is your quadratic equation in terms of a

[iggymiggy]

Thanks Factor30, some pointers, but I still don't get it.

I still can't see how to get to the second line.

[factor-fiction]

Which second line? Do you mean when solving the final quadratic?

[StBruno]

To date12a^2=4a^2+4b^2+8ab
So div by4 we get 3a^2=a^2+b^2+2ab
So 3a^2=(a+b)^2
So a x sqrt(3) =a+b
So a x (sqrt(3) - 1) =b

[factor-fiction]

Yes that's an alternative and maybe easier way

[factor-fiction]

Sorry- I didn't really look at that properly Stbruno- I just saw that you did it without using the formula.
I'll look after tea

[vascop]

Surely the easiest way is as follows:
Since (2a X Sqrt(2))^2 + 4a^2 =12a^2 then
12a^2=(2a+2b)^2 and so
+/-2 x sqrt(3)=2a+2b (notice there are 2 solutions)and so the 2 solutions are
2b=2a(sqrt(3)-1) and 2b=-2a(sqrt(3)+1)

If a is positive then the first answer for b is positive and the second negative and vice versa.

[vascop]

By the way +/- stands for 'plus or minus' which is usually written as a plus sitting on top of a minus.