Find all possible values of 𝑥 satisfying 𝑓 of 𝑥 is equal to 𝑥 squared plus nine 𝑥 plus two and 𝑓 of 𝑥 is equal to negative 16, given that 𝑥 is in the set of integers.
We’re given this quadratic equation 𝑓 of 𝑥 equals 𝑥 squared plus nine 𝑥 plus two. And we’re trying to answer the question, for what 𝑥-values will this function be equal to negative 16? If we’re going to solve this algebraically, we take our function 𝑓 of 𝑥 is equal to 𝑥 squared plus nine 𝑥 plus two. And in place of 𝑓 of 𝑥, we’re going to substitute negative 16 so that we have a new equation that says negative 16 is equal to 𝑥 squared plus nine 𝑥 plus two.
But to solve a quadratic like this, we want to set it equal to zero. And we can do that by adding 16 to both sides of this equation. Now we have zero equals 𝑥 squared plus nine 𝑥 plus 18. It seems like we can probably solve this by factoring. So we break up our 𝑥 squared into 𝑥 times 𝑥. And then we’re looking for two values that when multiplied together equal positive 18 and when added together equal nine. We have three times six, which multiply together to equal 18 and add together to equal nine. This means 𝑥 squared plus nine 𝑥 plus 18 can be broken up into the two terms 𝑥 plus three times 𝑥 plus six.
To solve for 𝑥, we set both of these equations equal to zero. On the left, we subtract three from both sides, and we see that 𝑥 equals negative three. On the right, we subtract six from both sides, and we see that 𝑥 equals negative six. If we wanted to check that we have found the correct values for 𝑥 that satisfy this equation, we would want to check that negative 16 is equal to negative three squared plus nine times negative three plus two. We get nine minus 27 plus two, which is equal to negative 16. And that verifies negative three as a solution.
We’ll use the same procedure to check for negative six. Negative 16 is equal to negative six squared plus nine times negative six plus two. We’ll get 36 minus 54 plus two, which is negative 16 and therefore verifies the 𝑥 equals negative six solution. It’s also true that negative three and negative six are integers. So we can say all possible values of 𝑥 are negative three and negative six.