my friend brought me this solution and i couldn't find the mistake. can you help me?

Question

1=1
(5-4)^2=(3-4)^2
taking square root in both sides
5-4=3-4
(ading 4 on both sides)
3=5 !!!!!!!

(how can 3 equal 5)

Answer 1

When you take the square root of anything, the answer can be plus or minus. When you chose the answers you did not match plus with plus or minus with minus.
Thus 5-4= -(3-4). That is where you goofed up.

Answer 2

Here's another way 3 = 5:
3 + 2 × 2 - 2 = 3 + 2 × 2 - 2
3 + 2 × 0 = 3 + 4 - 2
3 + 0 = 7 - 2
3 = 5.

Remember in first year algebra or pre-algebra when you learned of the proper order of operations for evaluating expressions? You can probably see the error in mine, so why's it so hard to see it in yours? Exponents and roots must be evaluated together... AFTER expressions inside grouping symbols.

√(5 - 4)² = √(3 - 4)²
√(1)² = √(-1)²
√(1) = √(1)
1 = 1

The reason your so-called "proof" is invalid is because it violates the very rules from which algebraic principles are founded.

"Grouping symbols," for further reference, include parentheses, brackets, braces, the fraction bar, and the radical. (You know this is true because when solving problems using the quadratic formula, you've always fully evaluated the expression under the radical before finding your solutions for the variable.)

For NORMAL and rfamilymember: The square root of a number is never "plus or minus." √1 = 1, √ 4 = 2, √8.41 = 2.9, and so on. Square roots of numbers are ALWAYS non-negative. That's why when you enter them in a calculator, you always get a non-negative answer. You two are confusing the thought of evaluating an expression (without variables) with solving an equation (which does have them). A quadratic equation such as x² = 1 has two solutions: x = ±1, but the expression √1 has only one simplification... it's 1.

Answer 3

First... What is discussed in the problem is absolute value. Absolute value, by defintion, is sqrt(x)^2. Absolute value is basically the distance that a number is from zero on a one-dimensional number line.

what the first step is really saying is...
sqrt(5-4)^2 = sqrt(3-4)^2 or |5-4| = |3-4|

5-4=1 and 3-4= (-1). The absolute value of 1 is 1, and the absolute value of (-1) is 1. So the two sides of the equation are equal. However this does not in any way imply that 3=5. It only means that the absolute values are the same.

Answer 4

(5 - 4)² = (3 - 4)² is true
Common sense tells us that
5 - 4 = 3 - 4 is not true

Therefore this brings us to thinking that the mistake is between these steps.
Our experience makes us realize that a number has two square roots (the positive and the negative one). although these numbers are not equal, their squares are equal. Take note that powers often bring the equation into an untrue statement. You should be very careful when to square or square root.

^_^

Answer 5

any square root has 2 solutions.

a^2 =+a or -a

so
on root [ (5-4)^2 ] = 5-4 or 4-5
same for 3-5

the correct answer however has to be logical
so as 3 never equals 5 LOGICALLY!!

it has to be 4-3 on RHS
giving 1=1

Answer 6

Here's another way 3 = 5:
3 + 2 × 2 - 2 = 3 + 2 × 2 - 2
3 + 2 × 0 = 3 + 4 - 2
3 + 0 = 7 - 2
3 = 5.

Remember in first year algebra or pre-algebra when you learned of the proper order of operations for evaluating expressions? You can probably see the error in mine, so why's it so hard to see it in yours? Exponents and roots must be evaluated together... AFTER expressions inside grouping symbols.

√(5 - 4)² = √(3 - 4)²
√(1)² = √(-1)²
√(1) = √(1)
1 = 1

The reason your so-called "proof" is invalid is because it violates the very rules from which algebraic principles are founded.

"Grouping symbols," for further reference, include parentheses, brackets, braces, the fraction bar, and the radical. (You know this is true because when solving problems using the quadratic formula, you've always fully evaluated the expression under the radical before finding your solutions for the variable.)

Answer 7

till 2nd step its right ie
(5-4)^2=(3-4)^2

but when we take square root of any number there are 2 possibilities that r the no. may be negative or it may be positive. hence solution further is as below...

(5-4) or -(5-4)=(3-4) or -(3-4)

adding 4 on both side:

5 or 3 = 3 or 5

now any idiot can tell that
5=5 and 3=3

hope ur not an idiot.

Answer 8

Just remember Please Excuse My Dear Aunt Sally (PEMDAS)
parentheses, exponents, multiplication, division, addition, subtraction.

You have to get ride of the parentheses first before you can take the square root of both sides!!!

(5-4)^2 = (1)^2 = 1
(3-4)^2 = (-1)^2 = 1 ( remember two negative number multiplied together equal a positive)
1= 1

Answer 9

You can't square root both sides at the same time. Square rooting both sides DOES NOT mean they are still equal. This is because squares of negative are positive, so if you square root two sides that are positive, one side might be negative and the other side positive. Negative does not equal to positive. Learn your maths properly before asking.

Answer 10

Ya, you have to keep in mind that a square root, or any even root of a value gives both + and - answers (abs value). Hence, +-1 = +-1 . I wish I had the 'plus minus' sign because writing +-1 like that is wrong. The plus should be on top of the - to signify the plus/minus expression.

Answer 11

Remember your order of operations.
(5-4)^2=(3-4)^2
Do the math in the parenthasis first
1^2=-1^2
1 x 1= -1 x -1
1=1
I hope this helps.