I have gathered here a collection of mistakes that are pretty easy to make.
Try to avoid these!
|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−5−a|
And be careful of these ones too:
|xx+y = xx + xy|
We can't simplify that!
Imagine x=4 and y=5:
44+5 = 49
That is definitely not equal to 44 + 45 (which actually equals more than 1)
Maybe you were thinking of this kind of fraction that we can simplify:
|x+yx = xx + yx|
Square root of xy
√(xy) =√x√y ... but not always!
Example: x = −5 and y = −2
So, does √10 = −√10 ??? I think not!
√(xy) =√x√y only when x and y are both >= 0
Two Equals One
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.
Example: Solve x2 – 5x = 2
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