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Video: Manipulating Quadratic Expressions by Completing the Square

Bethani Gasparine

Given that 3𝑥² + 3𝑥 + 5 = 𝑎(𝑥 + 𝑝)² + 𝑞, what are the values of 𝑎, 𝑝, and 𝑞?

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Video Transcript

Given that three π‘₯ squared plus three π‘₯ plus five is equal to π‘Ž times π‘₯ plus 𝑝 squared plus π‘ž, what are the values of π‘Ž, 𝑝, and π‘ž?

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 π‘₯ squared plus π‘₯ 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 π‘₯ 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 π‘Ž times π‘₯ plus 𝑝 squared plus π‘ž, π‘Ž would be three, 𝑝 would be one-half, and π‘ž would be seventeen-fourths. So again, π‘Ž equals three, 𝑝 equals one-half, and π‘ž equals seventeen-fourths.