About this video
The quadratic x^2 + 5x + c is a perfect square equal to (x + d)^2, with c and d positive rational numbers. Sal expands (x + d)^2 = x^2 + 2dx + d^2 and matches terms: 2d = 5 so d = 5/2, and c = d^2 = 25/4.
You'll be able to
- expand (x + d)^2 as x^2 + 2dx + d^2.
- match coefficients to find unknown values in a perfect square.
- find c and d when x^2 + bx + c = (x + d)^2.
Key words
- perfect square
- An expression that equals a binomial times itself, like (x + d)^2.
- rational number
- A number that can be written as a fraction, like 5/2.
- coefficient
- The number multiplying a variable, like 5 in 5x.
Transcript
- [Voiceover] The quadratic expression x-squared plus five x plus c is a perfect square. It can be factored as x plus d-squared. Both c and d are positive rational numbers. What I wanna figure out in this video is what is c, given the information that we have right over here?
What is c going to be equal to? And what is d going to be equal to? Like always, pause the video and see if you can figure it out. Let's work through this together. We're saying that x-squared plus five x plus c can be rewritten as x plus d-squared. Let me write that down.
So this part, this part, x-squared plus five x plus c, we're saying that, that could be written as x plus d-squared. This is equal to x plus d-squared. Now we can rewrite, x plus d-squared is going to be equal to x-squared plus two dx plus d-squared. If this step, right over here, you find strange, I encourage you to watch the videos on squaring binomials or on perfect square polynomials, either one, so you can see the pattern that this is going to be.
X squared plus two times the product of both of these terms plus d-squared. When you look at it like this, you can start to pattern match a little bit. You can say, alright, five x, right over here, that is going to have to be equal to two d, and then, you can also say, that c is going to have to be equal to d-squared.
Once again, you can say two d is equal to five, two d is equal to five, or that d is equal to five halves. We've figured out what d is equal to. Now we can figure out what c is, because we know that c needs to be equal to d-squared, gimme that orange color, actually, so we know that c is equal to d-squared, which is the same thing as five halves, squared.
We just figured out what d is equal to. Gonna be five halves, squared, which is going to be 25 over four. C is equal to 25 over four, d is equal to five halves. We're done.
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