# Algebra Mistakes

We 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 |

And be careful of these ones too:

## Simplifying Fractions

\frac{x}{x+y} = \frac{x}{x} + \frac{x}{y} |

We can't simplify that!

Imagine x=4 and y=5:

\frac{4}{4+5} = \frac{4}{9}

That is definitely not equal to \frac{4}{4} + \frac{4}{5} (which actually equals more than 1)

Maybe you were thinking of this kind of fraction that we **can** simplify:

\frac{x+y}{x} = \frac{x}{x} + \frac{y}{x} |

## Square root of xy

**√(xy) =√x√y** ... but not always!

### Example: x = −5 and y = −2

*(the mistake)*

^{2}√5√2

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

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

## Two Equals One

### Example:

^{2}= xy

^{2}from both sides: x

^{2}− y

^{2}= xy − y

^{2}

*(the mistake)*

**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 x^{2} – 5x = 2

^{2}– 5x = 2

*(the mistake)*

Let's check x=2:

2^{2} – 5×2 = 4−10 = **−6**, but we wanted x^{2} – 5x = **2**

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