Given that three 𝑥 squared plus three 𝑥 plus five is equal to 𝑎 times 𝑥 plus 𝑝 squared plus 𝑞, what are the values of 𝑎, 𝑝, and 𝑞?
Essentially, we need to take our function and complete the square. So the first thing that we need to do is to group the first two terms together, and now take out a greatest common factor, excluding the variables.
And now our goal is to turn this into something that is squared, so we need to make this a trinomial. So the first thing that we need to do is to take 𝑏 over two and square it. And in this case, 𝑏 is one, and one-half squared is one-fourth. So we can plug in one-fourth.
However, we can’t just add a number to our function. So if we’re adding one-fourth, we need to subtract it from the end. However, that isn’t actually a one-fourth; that is actually a value of three times one-fourth, which is three-fourths. So we need to subtract three-fourths from the edge, so we have three times 𝑥 squared plus 𝑥 plus one-fourth plus.
To add our fractions, I need to have a common denominator. And five is the same as 20 over four, so twenty-fourths minus three-fourths is equal to seventeen-fourths.
Now like we said before, we wanted what’s in the parentheses to be something squared, and it is. That is actually equal to 𝑥 plus one-half squared, because one-half times one-half is equal to one-fourth, and then we add one-half and one-half together; we get one. So 𝑏 is one and 𝑐 is the one-fourth.
So if this is equal to 𝑎 times 𝑥 plus 𝑝 squared plus 𝑞, 𝑎 would be three, 𝑝 would be one-half, and 𝑞 would be seventeen-fourths. So again, 𝑎 equals three, 𝑝 equals one-half, and 𝑞 equals seventeen-fourths.