Can anyone tell me what's wrong with this proof?

Question

Start:
-8 = -8
4 - 12 = 16 - 24
4 - (6 + 6) = 16 - (12 + 12)
4 - (6 + 6) + 9 = 16 - (12 + 12) + 9
factor:
(2 - 3)² = (4 - 3)²
2 - 3 = 4 - 3
2 = 4
divide by 2
1 = 2
Add 1:
1 + 1 = 3

What's wrong with this proof?
Also does a^1/2 = b^1/2 imply a=b ?

Answer 1

Forget all of the steps and go to the line after the word factor.

This line states (-1)^2 = (1)^2

which is clearly true.

However, it does not imply that -1 = 1, because taking the square root on both sides does not preserve the equality. Think of it this way - the square root of 4 is + or - 2, not 2. So 4 = 4 does not imply sqrt (4) = sqrt(4) because of the way the square root operation works.

Answer 2

Since -1^2 =1^2, there is no point in the first 4 lines as you could have started from there itself and gone on.
what you have done is you have used some special properties of some unique numbers (in your case the number 1, and the second power) which means that the suppositions do not hold true for any other number or any other power.
another example would be of 0 being used to prove :
4 x 0 = 5 x 0
therefore 4 = 5.

Answer 3

1. While taking square root consider both positive and negative values. Take the positive value of the first factor (2-3) and the negative value of the second factor (4-3) and equate it. That's how we should equate. Explained Below...
(2-3)^2=(4-3)^2
-(2-3)=+(4-3) [or] (2-3)=-(4-3)
If you equate like this you won't get that incorrect.
2. The same concept is applied to the second question.

Answer 4

when you factored out the exponents ^2 you didn't do the math

(2-3)^2 = (-1)^2 = -1/-1 = 1
(4-3)^2 = (-1)^2 = -1/-1 = 1

So where you're left with 2 - 3 = 4 - 3 you should have had -1/-1 = -1/-1 or 1 = 1

Answer 5

When you root the (2-3)^2 = (4-3)^2
You ignore the possibility that for
a^2 = b^2
a can equal -b
you consider the more usual a=b ,(and -a=-b => a=b) which isn't the case here.
I suppose this answers the other part of your question too.
a^1/2 = b^1/2 does imply a=b but not necessarily vice-versa.

Answer 6

(2 - 3)² = (4 - 3)²
2 - 3 = 4 - 3 This should be

(2 - 3)² = (4 - 3)²
2 - 3 = + or - (4-3)
Only the minus is alright

Th

Answer 7

a^2 = b^2 means a=b or a = -b
(2-3) is -(4-3) and not 4-3
that is the wrong step

Answer 8

(2-3)²=(4-3)²
but (2-3) is not equal to (4-3) because according to laws of indices if 2 like terms sit beside each other the simplification should be done there.
so,

(2-3)²=(4-3)²
.
. . (-1)²=(1)²
1=1 Q.E.D


But u have this medhod which have no mistakes like u have done.


(x+3)²=(x-2)² (Note:-There are no like terms,This is the trick of avoiding the mistake u have done.To avoid such mistakes i have taken x)
x+3=x-2
3=-2

Where's the mistake here ???????? If any please point out.

Answer 9

root of (2-3)^2 is not equal to 2-3
because root of (2-3)^2 = root of (-1)^2 which is wrong
u cant square root the square of two negative numbers

Answer 10

Your substitution at the very beginning is arbitrary. Hence the mistake

(a-b)² = a²-2ab+b², is a simple quadratic equation.

However,
(a-b)² not= (c-b)², if a not= c

Since 4 is not equal to 16, you cannot make this equation at step2 itself.