Question

How do you factor completely `x^2-12x+35`?

Answer 1

`(x-5)(x-7)=>x=5, x=7`

Explanation:

We need to find two numbers that add up to `-12` and those same two numbers have a product of `35`. After some thinking and trial and error, we come up with

`-5+ -7=-12` & `-5*-7=35`

Therefore `-5` and `-7` will be our two factors. We have:

`(x-5)(x-7)=0`

as our factored form, and to find the zeros, take the opposite sign to get:

`x=5` and `x=7`

Answer 2

`(x-7)(x-5)`

Explanation:

`x^2 - 12x + 35`

So what my math teacher used to teach me is that the place where the `35` is, that's where numbers multiply and their product is `35`. Where the `-12` is, that's where numbers add up and its result is `-12`.

However, the numbers have to be the same, so what do we do?

We first list all the factors of `35`:

• `1 * 35`
• `5 * 7`
• `7 * 5`
• `35 * 1`

And then the negatives (wink-wink it's going to be negatives...)

• `-1 * -35`
• `-5 * -7`
• `-7 * -5`
• `-35 * -1`

And then the one negative one positive... and so on. However, did you noticed that the middle number is negative?

A positive times positive can only have a positive result. That is eliminated. If it is a negative time a positive, then it can only be:

• `-35 + 1 = -34`
• `-1 + 35 = 34`
• `-7 + 5 = -2`
• `-5 + 7 = 2`

That are the only possible answers. But none of them are `-12`.
So the only way is negative times a negative. And the only combination that results in a `-12` is: `-7` and `-5`. Added together is `-12`. Multiplied together is `35`.

Voila, you have your answer. And of course, you put it with the `x` also; don't forget that!!

Answer: `(x-7)(x-5)`

Double check:

• `x * x`
• `-7x -5x`
• `-7 * -5`

so

`x^2 - 12x + 35`

Hope this helps!! :)

Answer 3

Find the 2 factors of 35 that also add to -12

Explanation:

Because our equation is already in the form `ax^2+bx+c`, we are ready to begin factoring.

Begin by finding the factors that multiply to 35 and add to -12.

Personally, if I don't see the factors after a few seconds I will begin listing ALL the factors of the "c" term and begin fiddling with the signs of the individual factors until I find something that works.

Factors: `(1, 35), (5,7)` are our only factors. We know that 5 and 7 multiply to 35 and add to 12. But, if we take the negative versions of both of these factors it still multiplies to 35 and adds to -12. These are our factors.

We then take those factors and rewrite our equation in the following form

`(x-5)*(x-7)`

These are the factors that you will be looking for. To check our work, simply distribute to undo the factoring process.