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

Mar 17, 2018

$\left(x - 5\right) \left(x - 7\right) \implies x = 5 , x = 7$

#### 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

$- 5 + - 7 = - 12$ & $- 5 \cdot - 7 = 35$

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

$\left(x - 5\right) \left(x - 7\right) = 0$

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

$x = 5$ and $x = 7$

Mar 17, 2018

$\left(x - 7\right) \left(x - 5\right)$

#### Explanation:

${x}^{2} - 12 x + 35$

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 $35$:

• $1 \cdot 35$
• $5 \cdot 7$
• $7 \cdot 5$
• $35 \cdot 1$

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

• $- 1 \cdot - 35$
• $- 5 \cdot - 7$
• $- 7 \cdot - 5$
• $- 35 \cdot - 1$

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: $\left(x - 7\right) \left(x - 5\right)$

Double check:

• $x \cdot x$
• $- 7 x - 5 x$
• $- 7 \cdot - 5$

so

${x}^{2} - 12 x + 35$

Hope this helps!! :)

Mar 17, 2018

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

#### Explanation:

Because our equation is already in the form $a {x}^{2} + b x + c$, 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: $\left(1 , 35\right) , \left(5 , 7\right)$ 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

$\left(x - 5\right) \cdot \left(x - 7\right)$

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