About this video
The instructor factors 2x^2 + 16x + 30 by first pulling out a common factor of 2, leaving x^2 + 8x + 15 inside. The key step is finding two numbers that multiply to 15 and add to 8, which gives b + c = 8, with b and c equal to 3 and 5 in either order. Expanding the factored binomials back out confirms the result.
You'll be able to
- factor out a common factor before factoring a quadratic.
- find two numbers whose product and sum match the coefficients of a trinomial.
- verify a factorization by multiplying the binomials back out.
Key words
- Quadratic
- A polynomial whose highest power of the variable is squared.
- Common factor
- A number or term that divides every term of an expression.
- Factoring
- Rewriting an expression as a product of simpler expressions.
- Binomial
- An expression with two terms, such as (x + 3).
- Sum and product
- For a trinomial x^2 + bx + c, the numbers b + c and b times c must match the middle and last coefficients' pattern.
Transcript
- [Instructor] We're asked in the equation above, b and c are constants. What is the value of b + c? And they give us the equation over here. So, pause this video and see if you can have a go at that before we work through this together. All right, now let's work through this together.
And it looks like what's happening is we have a quadratic on the left and then on the right we have that same quadratic that is factored out, although they don't tell us what b and c are we have to figure that out. So, one way to tackle this is actually let me just rewrite the left-hand side of this.
So, it is 2x squared + 16x + 30. And what I wanna do is try to get as close to the form that I have on the right as possible. So, it looks like they factored out a two. So, let me do that. So, this is equal to two times. And if any of this factoring of quadratics is unfamiliar to you, I encourage you to review that on Khan Academy, on The non-ACT portion of Khan Academy to get the basics.
But if we factor out of two out of this first term, you're just left with an X squared. You factor out of two out of 16x, you get 8x. And you factor a two out of 30 and you get + 15. And then it looks like what they have done is they have factored this part into x + b X x + c.
And the simplest way to factor things is to say, all right, are there two numbers that when I add them, I get eight, and that when I multiply them, I get 15? And those two numbers are actually going to be b and c this is one of our main factoring techniques. So, b + c needs to be equal to eight, and b X c needs to be equal to 15.
And if we figure that out, then we can factor completely. Well, we've just actually answered their question, b + c needs to be equal to eight. And so, eight is the answer. Now, let me just factor this out completely, so that you can see that a little bit more completely.
I'm using the word completely a lot. So, if I were to factor this out, this is the same thing as two times the two numbers that add up to eight. And when I multiply them, I get 15, let's see three and five seemed to work. So, it's gonna be two times X plus three times X plus five.
You can verify that three times five you're gonna get that 15 there. And then when you multiply these two binomials, you're gonna get 3x + 5x, which is going to be 8x. And so, you can see that you can either treat b as three and c is five or b is five and c is three, but either way, b + c is going to be equal to eight.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.