Algebra Mistakes

I have gathered here a collection of mistakes that are pretty easy to make.

Try to avoid these!

Mistake
not
Correction
yes
x2 = 25, so x = 5 x = 5 or x = −5
(x−5)2 = x2 − 25 = (x−5)(x−5) = x2 − 10x + 25
√(x2+y2) = x + y √(x2+y2) is as far as we can go
   
x2x4 = x8 = x6 (add exponents)
(x2)4 = x6 = x8 (multiply exponents)
2x-1 = 1/(2x) = 2/x
−52 = 25 = 25 (do exponent before minus)
(−5)2 = −25 = +25 (do brackets before exponent)
5½ = 1/52 = √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−5a
   
not yes

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
 =i2√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: x2 = xy
Subtract y2 from both sides: x2 − y2 = xy − y2
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 x2 – 5x = 2

Start with:x2 – 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:

22 – 5×2 = 4−10 = −6, but we wanted x2 – 5x = 2

That only works when x(x−5) = 0 (zero) not any other number