2020/02/15

Let’s see next formula.

\[ 6x^2 + 23x - 48 \]

If you are asked to factorize this formula, how will you do it? Because this is quadratic equation, it will be this form.

\[ (ax + b)(cx + d) \]

Now, we have to know what a, b, c and d are. This is the result of factorization of above formula.

\[ (3x + 16)(2x - 3) \]

There are some ways to solve this problem, but one popular way is writing this diagram.

```
3 16
×
2 -3
```

Multiply 3 and 2 is 6. Multiply 16 and -3 is -48.

Now, 2 numbers arranged diagonally (3 and -3, 2 and 16) can represent 23.

\[ 3 * -3 + 2 * 16 = 23 \]

To do factorization in this way, the steps are like this:

- Write 2 numbers which becomes a coefficint of \(x^2\) (6) if they are multiplied
- Write 2 numbers which becomes constant term (-48) if they are multiplied
- If the result of the sum of diagonal multiplication (\(3 * -3 + 2 * 16 = 23\)) is the same as a coefficint of \(x\) (23), then the answer is found.
- If they are different, try another multiplication

The problem of this solution is that we are not sure how many times we have to try. For example, divisors of 6 are 1, 2, 3, 6. So, this diagram can be this 2 patterns:

```
6
1
```

or

```
3
2
```

48 is \(2^4 * 3\) . So, it has more patterns. \((1, 48)\), \((2, 24)\), \((3, 16)\), \((4, 12)\), \((6, 8)\).

Because of above, this diagram can be 10 patterns (\(2 * 5\)). Also, we have to take care of - (minus). It becomes more. But we can simplify this method.

To get straight to the point, we can find the answer by 2 times trial at most.

First, we need to check (6, 1) can be an answer.

```
6
1
```

Now, we don’t need to check all the cases for -48. Only possible answer is (1, 48).

```
6 +-1
1 -+48
```

We can ignore other cases like (2, 24). Why?

If (6, 1) and (2, 24) works, factorized answer will be like this:

\[ (6x \pm 2)(x \mp 24) \]

But, this can be still factorized:

\[ 2(3x \pm 1)(x \mp 24) \]

If this can be an answer, given formula must be able to be factorized by 2. However, it’s obviously impossible.
In other words, left number and right number must be *coprime*. In this case, 6 and 2 is not coprime. So it never gets an answer.

```
6 +-1
1 -+48
```

This also looks not leading us to an answer. So, let’s see next case (3, 2).

```
3
2
```

48 is composed of \(2 * 2 * 2 * 2 * 3\). 6 is \(2 * 3\). To make them coprime, only possible answer is (16, 3).

```
3 +-16
2 -+3
```

Now, we only need to take care about sign.