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How to Solve a Quadratic Equation by Factoring

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  • 0:05 Solve with Factoring
  • 2:35 Example #1
  • 4:47 Example #2
  • 7:16 Lesson Summary
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Lesson Transcript
Instructor: Luke Winspur

Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.

If your favorite video game, 'Furious Fowls,' gave you the quadratic equation for each shot you made, would you be able to solve the equation to make sure every one hit its target? If not, you will after watching this video!

Solve a Quadratic Equation with Factoring

Welcome to level one of 'Furious Fowls,' the game that puts you in control of the birds that are trying to get their eggs back from those pesky pigs. Can you launch your bird at the exact right angle to make sure your bird gets a direct hit? We're about to find out.

The only difference between 'Furious Fowls' and other games out there that are similar is that you only get one shot to make your hit. So it better be worth it! But, because you only get one shot, we will give you a hint to help you out and make sure your first shot is always a direct hit. What kind of a hint am I talking about? Let's check out level one.

So here we go; our bird's all set in our sling shot ready to go, and our pig is waiting very smugly over there, thinking it's safe. But, we want to make sure we get him. I can try to eyeball it by swinging my bird up and down here. Notice that as I swing it in different directions, the mathematical equation in the corner of the screen is changing. It's going to be your job to use that mathematical equation to make sure that your one shot hits its target.

Factoring changes the equation from standard to intercept form.
solve using factoring method

Example #1

Let's go ahead and try to line up a shot. Maybe like this; that looks about right. The game has told us that if we fire at this angle, our bird will fly in a path that follows the quadratic equation y = -x2 + 6x + 16 where y is the height of the bird and x is the distance it has traveled. Since we're trying to hit a pig that's sitting on the ground, we're curious when our bird will hit the ground, which is where y = 0. Substituting this value in gives us this equation. This equation is asking us the question, how far will our bird go before it hits the ground?

We normally solve equations by using inverse operations to get the variable by itself, but in this case, we'd run into some problems with that strategy. Because there are xs but also x2's in this equation, there isn't just one variable we can get by itself. This means we need a new strategy for solving quadratic equations. While there are a few different ways to do this, the most basic is through factoring. While this way won't always work, it is probably the easiest way to do it, so knowing how to is really helpful. Factoring the quadratic will change it from standard form into intercept form, which will tell us exactly where the parabola crosses the x-axis (where its x intercepts are), or in this problem, where the bird hits the ground.

To factor a quadratic, it's really nice not having a negative x2 term, so let's divide that part out first. Pulling a negative from each term leaves us with this equation instead. Now we can begin to look for the two numbers that have a product of -16 and a sum of -6. The reason we're doing that is all about factoring. So if you're struggling with factoring, you should probably go back and check out a previous lesson to get the basics. Finding a pair of numbers that satisfy these two equations will be easiest if we write out the different factors of 16 and look for the pair of them that can add or subtract to -6. Doing that makes it look like 2 and -8 are our winners. We can therefore rewrite our equation as this: (0 = -(x - 8)(x + 2)).

But why did we do that? How has that gotten us any closer to solving this equation for x? I still have two of them; I can't get one of them by themselves. Well, what we now have is a product that equals zero. This allows us to use what is called the zero product property, which says that anytime a product equals zero, at least one of the things we are multiplying must be zero. Another way of saying that is this: the only way to multiply two things and end up with zero is by having one of those two things be zero itself. What this allows us to do is split the equation up and say that either x - 8 or x + 2 is equal to zero. Now that the xs are separated, we can again use inverse operations to solve for them. Undoing -8 and +2 tells us that the two places where our bird will hit the ground are at x = -2 or x = 8.

Well, in our game here that -2 answer might happen if the slingshot broke, but we're more concerned with the answer of 8. And it looks now, if our bird flies out to 8, it will go right over the pig we're trying to hit, so making this shot would be a waste of a bird.

Example #2

Use the area method to find the factored expression in example #2.
Factoring Quadratic Area Method

But if we slightly adjust it, maybe aiming up a little bit so that our bird flies higher in the air but doesn't go as far, we are told it will now fly the path of this parabola y = -2x2 + 11x + 21. Let's see if this does what we want it to.

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