Solve the equation 𝑥 squared minus eight 𝑥 plus 16 equals zero by factoring.
Immediately, we can notice that our a term or our leading coefficient is one. This means the equation will be a little bit simpler to factor. It means that our first term in both of our factors will be 𝑥.
We just need to work to figure out what our second term would be. We do that by finding factors of 16 then add up to negative eight. One and 16 are factors of 16. Two and eight are factors of 16. Three is not a factor of 16. But four is a factor, because four times four equals 16.
Now we need to determine which combination of these two factors when added together would produce negative eight. So that means they would have to add together to equal negative eight. But not only do they have to add up to negative eight, they also have to multiply together to equal 16, positive 16. One plus 16 does not equal negative eight.
We can try adding negative one plus negative 16. We can do this because negative one times negative 16 still equals positive 16. But no matter what we try, one and 16 don’t add up to negative eight. And as I look at two and eight, I also realized they will not add up to equal negative eight. But something is happening in this third line. Negative four plus negative four does equal negative eight, and negative four times negative four equals positive 16.
We found our two terms. Each term is negative four here: 𝑥 minus four times 𝑥 minus four. We can’t forget to bring down our zero. We also want to notice that our question didn’t ask us to factor this problem. It asked us to solve by factoring. When we solve these equations, the final answer will be 𝑥 equals something. In order for this equation to be zero, one of the factors must be zero.
We set each of the two factors equal to zero; 𝑥 minus four equals zero. And because they’re the same, we have 𝑥 minus four equal to zero again. We can solve this equation by adding four to both sides, and we find that 𝑥 equals four. And since they are the same exact factor repeated twice, they both equal four. That means there’s only one solution. There’s only one place where this equation is equal to zero, and that’s when 𝑥 equals four.