Algebra Mistakes
I have gathered here a collection of mistakes that are pretty easy to make.
Try to avoid these!
Mistake 
Correction 
x^{2} = 25, so x = 5 
x = 5 or x= 5 
(x5)^{2} = x^{2}  25 
= (x5)(x5) = x^{2}  10x + 25 
√(x^{2}+y^{2}) = x + y 
√(x^{2}+y^{2}) is as far as you can go 


x^{2}x^{4 }= x^{8}^{} 
^{ }= x^{6} (add exponents) 
(x^{2})^{4 }= x^{6}^{} 
^{ }= x^{8} (multiply exponents) 
2x^{1} = 1/(2x)^{} 
= 2/x 
5^{2} = 25 
= 25 (do exponent before minus) 
(5)^{2} = 25 
= +25 (do brackets before exponent) 
5^{½} = 1/5^{2} 
= √5 


log(a+b) = log(a) + log(b) 
log(a+b) is as far as you can go 


x(a/b) = xa/xb 
= xa/b 
x(5+a) = x5+a 
= x5a 
Here are some more mistakes in detail:
Square root of xy
√(xy) =√x√y ... but not always!
Example: x = 5 and y = 2
√10 
= √(5×2) 



= √(5)√(2) 

(The mistake) 

= i√5 × i√2 



= i^{2}√5√2 



=  √10 


So ... √10 =  √10 ??? I think not!
√(xy) =√x√y only when x and y are both >= 0
Two Equals One
Example:


x = y 


Multiply both sides by x: 

x^{2} = xy 


Subtract y^{2} from both sides: 

x^{2}  y^{2} = xy  y^{2} 


Factor: 

(x+y)(xy) = y(xy) 


Divide both sides by (xy): 

x + y = y 

(The mistake) 
Since x = y, we see that 

2y = y 


Thus 

2 = 1 


Why is that wrong? Silly person! You tried to divide by zero.
Remember we said that x=y, so (xy)=0 and going from (x+y)(xy) = y(xy) to x + y = y is a mistake.
Factoring
Example: Solve x^{2} – 5x = 2


x^{2} – 5x = 2 


Factor x: 

x(x5) = 2 


So: 

x=2 or x5=2 

(The mistake) 
Hence 

x=2 or 7 


That would only work when x(x5) = 0
