yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Solving quadratics by factoring: leading coefficient â   1 | High School Math | Khan Academy


3m read
·Nov 11, 2024

So we have (6x^2 - 120x + 600 = 0). Like always, pause this video and see if you can solve for (x). If you can find the (X) values that satisfy this equation.

All right, let's work through this together. So the numbers here don't seem like outlandish numbers; they seem like something that I might be able to deal with and I might be able to factor. So let's try to do that.

The first thing I like to do is see if I can get a coefficient of one on the second degree term on the (X^2) term. It looks like actually all of these terms are divisible by six. So if we divide both sides of this equation by six, I'm still going to have nice integer coefficients. So let's do that; let's divide both sides by six.

If we divide the left side by six, divide by six, divide by six, divide by six, and I divide the right side by six. If I do that, clearly, if I do the same thing to both sides of the equation, then the equality still holds. On the left-hand side, I am going to be left with (x^2), and then (-120 / 6) that is, let's see, (120) divided by (6) is (20), so that's (-20x).

Then (600) divided by (6) is (100), so plus (100) is equal to (0). Divided by (6) is equal to (0). So let's see if we can factor. If we can express this quadratic as the product of two expressions.

The way we think about this—and we've done it multiple times—is if we have something that is (x + a) times (x + b). This is hopefully a review for you; if you multiply that out, that is going to be equal to (x^2 + (a + b)x + ab).

So what we want to do is see if we can factor this into ( (x + a)(x + b) ). (A + B) needs to be equal to (-20) (that needs to be (a + b)), and then (a \times b) needs to be equal to the constant term (that needs to be (ab)).

So can we think of two numbers that, if we take their product, we get positive (100), and if we take their sum, we get (-20)? Well, since their product is positive, we know that they have the same sign. So they're both going to have the same sign; they're either both going to be positive, or they're both going to be negative.

Since we know that we have a positive product and since their sum is negative, well, they must both be negative. You can't add up two positive numbers and get a negative, so they both must be negative.

So let's think about it a little bit. What negative numbers, when I add them together, I get (-20), and when I multiply, I get (100)? Well, you could try to factor (100); you could say, well, (-2 \times -50) or (-4 \times -25), but the one that might jump out at you is (-10) times (-10).

And this is (-10 + -10), so in that case, both our (a) and our (b) would be (-10). We can rewrite the left side of this equation as ( (x - 10)(x - 10) ). Again, (x - 10) and that is going to be equal to zero.

All I've done is I've factored this quadratic, or another way, these are both the same thing as ( (x - 10)^2 = 0 ). So the only way that the left-hand side is going to be equal to zero is if (x - 10) is equal to zero.

You could think of this as taking the square root of both sides, and it doesn't matter if I take the positive or negative square root or both of them; it's the square root of (0).

So we would say that (x - 10) needs to be equal to zero, and so adding (10) to both sides, we have (x = 10) is the solution to this quadratic equation up here.

More Articles

View All
4.5 Billion Years in 1 Hour
Earth is 4.5 billion years old – impossible for your brain to truly grasp, so here is an experiment: every second, around 1.5 million years will pass – you’re on a musical train ride looking out the window, passing all of Earth’s history in an hour. Watch…
Separation of Powers and Checks and Balances
This is a great excerpt from Federalist 51 by James Madison. Just as a reminder, the Federalist Papers, which were written by Hamilton, Madison, and John Jay, were an attempt to get the Constitution passed, to get it ratified. So these were really kind of…
Indigenous Art in Canada | National Geographic
If you want to travel through indigenous country, experience the art. Whether it’s a painting, whether it’s a sculpture, whether it’s a song, every piece is the embodiment of a story. The art is the land, and the land is the art. This is how we share our …
Sound Meets Sculpture and Robotics - Tech+Art | Genius: Picasso
They say that every new technology has some potential military application, but I’d like to think that most new technologies seem to have musical possibilities and applications also. For about 300 years, the pipe organ was the most complex thing that huma…
Uncle Tom's Cabin part 2
So Becca and I have been talking about Uncle Tom’s Cabin, which is this book from the 1850s that Abraham Lincoln actually said started the Civil War. So how did this book start a war? In this video, we’ll tell you a little bit more about the plot. Um, bu…
Sin City's Deadly Vixen | Underworld, Inc.
One breed of hustler exploits this weakness to devastating effect. I’m about to go out and get me some money. I’m about to find some guy that wants to me, and I’m going to get him for all that he has. Vixen is a thief who POS uses as a prostitute. She get…