# 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 |

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) |
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= √(−5)√(−2) |
(The mistake) | ||

= i√5 × i√2 |
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= i^{2}√5√2 |
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= − √10 |

So, does **√10** = **− √10** ??? I think not!

**√(xy) =√x√y** *only when x and y are both >= 0*

## Two Equals One

### Example:

x = y |
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Multiply both sides by x: | x ^{2} = xy |
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Subtract y^{2} from both sides: |
x ^{2} − y^{2} = xy − y^{2} |
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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 |
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Thus | 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 x^{2} – 5x = 2

x ^{2} – 5x = 2 |
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Factor x: | x(x−5) = 2 |
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So: | x=2 or x−5=2 | (The mistake) |
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And so: | x=2 or 7 |

That only works when **x(x−5) = 0** (zero)