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 
(x−5)^{2} = x^{2} − 25  = (x−5)(x−5) = x^{2} − 10x + 25 
√(x^{2}+y^{2}) = x + y  √(x^{2}+y^{2}) is as far as we 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 we can go 
x(a/b) = xa/xb  = xa/b 
x−(5+a) = x−5+a  = x−5−a 
And be careful of these ones too:
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, does √10 = −√10 ??? I think not!
√(xy) =√x√y only when x and y are both >= 0
Two Equals One
Example:
Start with: 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)(x−y) = y(x−y)
Divide both sides by (x−y):x + y = y (the mistake)
Since x = y, we see that: 2y = y
And so: 2 = 1
Hang on! That can't be right!
What went wrong? Silly us! We tried to divide by zero.
When we said that x=y, it means that (x−y)=0 , so going from (x+y)(x−y) = y(x−y) to x + y = y is a mistake.
Factoring
Example: Solve x^{2} – 5x = 2
Start with:x^{2} – 5x = 2
Factor x:x(x−5) = 2
So:x=2 or x−5=2 (the mistake)
And so:x=2 or 7
Let's check x=2:
2^{2} – 5×2 = 4−10 = −6, but we wanted x^{2} – 5x = 2
That only works when x(x−5) = 0 (zero) not any other number