And now, it's time for a headache!

[Vicsun]

Please explain the following, because I failed to do so and it's driving me mad.

16 - 36 = 25 - 45 	//+81/4

16 - 36 + 81/4 = 25 - 45 + 81/4

4^2 - 2 * 4 * 9/2 + (9/2)^2 = 5^2 - 2 * 5 * 9/2 + (9/2)^2

4 - 9/2 = 5 - 9/2		//+9/2

4 = 5

[Lost One]

To me, it looks like the working out from the 3rd line to the 4th line makes no sense.

There's no way to explain it, except that the calculation is wrong. As both sides start equal, everything done to both will ultimately result in both still being equal.

[Vicsun]

Quote:

Originally Posted by Lost One To me, it looks like the working out from the 3rd line to the 4th line makes no sense.

It does.
(a + b)^2 = a^2 + 2ab + b^2
And what you have is basically a=4 b= -9/2 on the left side and a=5 b= -9/2 on the right side.

Between line 3 and line 4 one can insert:
(4 - 9/2)^2 = (5 - 9/2)^2 //sqrt

Quote:

There's no way to explain it, except that the calculation is wrong. As both sides start equal, everything done to both will ultimately result in both still being equal.

The calculation is indeed wrong, but I can't figure out what's wrong with it. That's why I posted here.

edit: I'm guessing the solution has got something to do with the fact that you can't square-root the above. But I'm still confused as to why.

[fable]

I've asked my wife to look at it. She does algorithms in her head to avoid long division, so she comes under the aegis of scary mathheads in my book. I'll post whatever she says.

EDIT: She doesn't know what the "//" stands for. Can you explain, please?

[Lost One]

Quote:

Originally Posted by fable EDIT: She doesn't know what the "//" stands for. Can you explain, please?

Took me a while to get that too. I thought it meant double division initially, but it's actually just a separator, and what comes after it, is what he's going to do to both sides of the equation.

Quote:

It does. (a + b)^2 = a^2 + 2ab + b^2 And what you have is basically a=4 b= -9/2 on the left side and a=5 b= -9/2 on the right side. Between line 3 and line 4 one can insert: (4 - 9/2)^2 = (5 - 9/2)^2 //sqrt

Hah, gotta love these tricky maths questions. Now that I understand better what you did, yes, you did go wrong, but it was actually before the square rooting. In fact, you were wrong at the third line.

When you made 16 into 4^2, 25 into 5^2, and 81/4 into 9/2....you forgot the minus from the equation.

16 is not equal to only (+4)^2. It is also equal to (-4)^2.

Thus, strictly speaking, your equation from the second line should have become:

(+-4)^2 - 2 * 4 * 9/2 + (+-9/2)^2 = (+-)5^2 - 2 * 5 * 9/2 + (+-9/2)^2

So, when you transform that into the form of (4 - 9/2)^2 = (5 - 9/2)^2, you are only taking positive values for a & b. And that is where you go wrong and why you couldn't square root the squared brackets.

[Vicsun]

Ahh... that makes sense. I will sleep comfortably tonight knowing that one of the greater mysteries plaguing my life has been solved.

[Rob-hin]

Train A leaves Chicago at 45 miles an hour blah die blah...

I truly hate math like this equasion and all math like it.

[Lost One]

While what I have said earlier is true, it doesn't stop the fact that using positive values should give a correct answer, if not a complete one.

After all, (4 - 9/2)^2 = (5 - 9/2)^2 is still true as both sides are equal to another.

Much like (1 - 2)^2 = (3 - 2)^2 and yet, if you square root both sides,

1 - 2 = 3 - 2 or 1 = 3

Clearly not true. This is because while you can square two sides to achieve the same answer, after your equation has become quadratic (you squared it, and now it has more than one solution) then you can't really attempt to prove your original equation backwards by square rooting both sides. Square rooting a quadratic equation always gives more than one answer.

so, +-sqrt(1 - 2)^2 = +-sqrt(3 - 2)^2

the answer is +-(1 - 2) = +-(3 - 2)

which gives, for +, -1 = +1 or +1 = -1....thus, the solution to this quadratic equation (by square rooting both sides) is +1, -1 (you cannot separate the -2s by adding 2 to each side). Note that the difference between the solutions +1 and -1 is equal to 2. Looking back to 1 = 3, 2 is exactly what differentiates them.

Going back to your equation, (4 - 9/2)^2 = (5 - 9/2)^2.

That should be the same as +-(4 - 9/2) = +-(5 - 9/2)

so, for + values, -1/2 = +1/2 and for - values, +1/2 = -1/2. The two solutions to your quadratic equation are +1/2, -1/2. The difference between each solution is 1. Looking back to 4 = 5, 1 is exactly what differentiates them.

So, my point? You cannot prove a single-solution equation by working backwards from a quadratic equation.

eg: 3 = 3..........3^2 = 3^2.........9 = 9

But sqrt(9) = +-3 (so you can't prove your first equation 3 = 3 by going backwards). But even so, there are ways to make sense out of it. In the meantime, I'm going to need a headache pill myself.