Algebra Mistakes
I 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 you 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 you can go |
| |
|
| x(a/b) = xa/xb |
= xa/b |
| x-(5+a) = x-5+a |
= x-5-a |
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 |
|
|
| |
= i2√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: |
|
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 |
|
|
| Thus |
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2 = 1 |
|
|
Why is that wrong? Silly person! You tried to divide by zero.
Remember we said that x=y, so (x-y)=0 and going from (x+y)(x-y) = y(x-y) to x + y = y is a mistake.
Factoring
Example: Solve x2 – 5x = 2
| |
|
x2 – 5x = 2 |
|
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| Factor x: |
|
x(x-5) = 2 |
|
|
| So: |
|
x=2 or x-5=2 |
|
(The mistake) |
| Hence |
|
x=2 or 7 |
|
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That would only work when x(x-5) = 0
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