Algebra Mistakes
We have gathered here a collection of mistakes that are pretty easy to make.
Try to avoid these!
Mistake |
Correction |
|---|---|
| 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 |
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And be careful of these ones too:
Simplifying Fractions
| 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
√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