# Basic Algebra/Rational Expressions and Equations/Adding and Subtracting When the Denominators are Different

## Lesson

If you have to add two rational fractions with different denominators, as the first step, you have to find the LCM:

```  3   +   3
x+1     x-1
```
```  LCM = (x+1)(x-1)
```

Now divide the LCM by both denominators and multiply by their respectives numerators:

```  (x+1)(x-1) / (x+1) = (x-1) . (3) = 3x-3
(x+1)(x-1) / (x-1) = (x+1) . (3) = 3x+3
```

The sum of the two results would be the new nominator:

```  3x-3+3x+3 =
(x+1)(x-1)
```
```     6x
(x+1)(x-1)
```

This is another example:

```  6x   +   9x
2x-6    x2-6x+9
```

We factorize both denominators and find the LCM

```  2x-6 = 2(x-3)
x2-6x+9 = (x-3)2
LCM = 2(x-3)2
```

Now we divide and multiply:

```  2(x-3)2 / 2(x-3) =
2x2-12x+18 / 2x-6 = x-3
(x-3) . 6x = 6x2-18x
```
```  2(x-3)2 / (x-3)2 =
2x2-12x+18 / x2-6x+9 = 2
(2) . (9x) = 18x
```

We add the results to obtain the nominator; the denominator is the LCM:

```  6x2-18x+18x =
2(x-3)2
```
```   6x2
2(x-3)2
```

We can factorize the nominator to simplify the result:

```  2(3x2) =
2(x-3)2
```
```  3x2
(x-3)2
```

Easy

Medium

Hard