Find, by factoring, the zeros of the function 𝑓 of 𝑥 equals 𝑥 squared plus two 𝑥 minus 35.
First, we can copy down our function. And then, we wanna know what do factoring means. It means we’ll take this 𝑥 squared and break it into two different sums. 𝑥 plus some number times 𝑥 plus some number will yield 𝑥 squared plus two 𝑥 minus 35.
The two missing values must multiply together to equal 35. They must be factors of 35. I know that one times 35 equals 35. That makes one and 35 factors. 35 is not divisible by two or three or four. It is divisible by five. Five times seven equals 35. That makes both five and seven factors of 35. 35 is not divisible by six.
And that means we found all the factors. But how do we decide between these two sets of factors? Is it 𝑥 plus one times 𝑥 plus 35 or is it 𝑥 plus five times 𝑥 plus seven? In the next step, we’ll need to look at two things: we’ll need to look at the signs and we’ll also need to look at the digit that represents variable 𝑏 — in our case, a two.
The first thing we see is that we’re dealing with negative 35, minus 35. And that means we would need to say plus one minus 35 or minus one plus 35. It also means we could say plus five minus seven or minus five plus seven. How do we go about narrowing these choices down? We need values that when they are added together, they equal positive two.
If we added negative one and positive 35, we will get 34. That’s not an option. Okay, so what if we change those signs? What if we have positive one and negative 35? Now, we have negative 34, still not an option. This means we can’t use these two factors. Down to five and seven, here we have negative five plus seven. Negative five plus seven is positive two.
What does that mean? It means that saying 𝑥 minus five times 𝑥 plus seven is the same thing as saying 𝑥 squared plus two 𝑥 minus 35. These functions are the same written in two different formats.
But back to the task at hand: we’re trying to find the zeros. We want to know when would 𝑓 of 𝑥 be equal to zero. Well, we know that zero times 𝑥 plus seven would be equal to zero or 𝑥 minus five times zero would be equal to zero. We need to know where both of these expressions would be equal to zero: when is 𝑥 minus five equals zero and when is 𝑥 plus seven equal to zero?
On the left, we add five to both sides of our equation. Negative five plus five cancels out. And 𝑥 is equal to positive five. On the right, we subtract seven from both sides. Plus seven minus seven cancels out. And 𝑥 is equal to negative seven.
The two places where this function would be equal to zero is when 𝑥 is equal to negative seven or five.