Solving quadratics by factoring
The height of a triangle is four inches less than the length of the base. The area of the triangle is 30 inches squared. Find the height and base. Use the formula area equals one half base times height for the area of a triangle. OK. So let's think about it a little bit. We have the-- let me draw a triangle here. So this is our triangle. And let's say that the length of this bottom side, that's the base, let's call that b. And then this is the height. This is the height right over here. And then the area is equal to one half base times height. Now in this first sentence they tell us at the height of a triangle is four inch is less than the length of the base. So the height is equal to the base minus 4. That's what that first sentence tells us. The area of the triangle is 30 inches squared. So if we take one half the base times the height we'll get 30 inches squared. Or we could say that 30 inches squared is equal to one half times the base, times the height. Now instead of putting an h in for height, we know that the height is the same thing as 4 less than the base. So let's put that in there. 4 less than the base. And then let's see what we get here. We get-- let me do this in yellow. We get 30 is equal to one half times-- let's distribute the b-- times b, let me make it clear. So let's do it this way. Times b over 2, times b, minus 4. I just multiplied the one half times the b. Now let's distribute the b over 2. So 30 is equal to b squared over 2, be careful. b over 2 times b is just b squared over 2. And then b over 2, times negative 4 is negative 2b. Now just to get rid of this fraction here let's multiply both sides of this equation by 2. So let's multiply that side by 2. And let's multiply that side by 2. On the left hand side you get 60. On the right hand side 2 times b squared over 2 is just b squared. Negative 2b times 2 is negative 4b. And now we have a quadratic here. And the best way to solve a quadratic-- we have a second degree term right here-- is to get all of the terms on one side of the equation, having them equal 0. So let's subtract 60 from both sides of this equation. And we get 0 equal to b squared, minus 4b, minus 60. And so what we need to do here is just factor this thing right now, or factor it. And then, no-- if I have the product of some things, and that equals 0, that means that either one or both of those things need to be equal to 0. So we need to factor b squared, minus 4b, minus 60. So what we want to do, we want to find two numbers whose sum is negative 4 and whose product is negative 60. Now, given that the product is negative, we know there are different signs. And this tells us that their absolute values are going to be four apart. That one is going to be four less than the others. So you could look at the products of the factors of 60. 1 and 60 are too far apart. Even if you made one of the negative, you would either get positive 59 as the sum or negative 59 as the sum. 2 and 30, still too far apart. 3 and 20, still too far apart. If you had made one negative you'd either get negative 17 or positive 17. Then you could have 4 and 15, still too far apart. If you made one of them negative, their sum would be either negative 11 or positive 11. Then you have 5 and a 12, still seems too far apart from each other. One of them is negative, then you either have their sum being positive 7 or negative 7. Then you have 6 and 10. Now this looks interesting. They are four apart. So if we make-- and we want the larger absolute magnitude number to be negative so that their sum is negative. So if we make it 6 and negative 10 their sum will be negative 4, and their product is negative 60. So that works. So you could literally say that this is equal to b plus 6, times b, minus 10. b plus the a, plus b minus the b. And let me be very careful here. This b over here, I want to make it very clear, is different than the b that we're using in the equation. I just used this b here to say, look, we're looking for two numbers that add up to this second term right over here. It's a different b. I could have said x plus y is equal to negative 4, and x times y is equal to negative 60. In fact, let me do it that way just so we don't get confused. So we could write x plus y is equal to negative 4. And then we have x times y is equal to negative 60. So we have b plus 6, times b plus y. x is 6, y is negative 10. And that is equal to 0. Let's just solve this right here. And then we'll go back and show you. You could also factor this by grouping. But just from this, we know that either one of these is equal to zero. Either b plus 6 is equal to 0, or b minus 10 is equal to 0. If we subtract 6 from both sides of this equation, we get b is equal to negative 6. Or if you add 10 to both sides of this equation, you get b is equal to 10. And those are our two solutions. You could put them back in and verify that they satisfy our constraints. Now the other way that you could solve this, and we're going to get exact same answer. Is you could just break up this negative 4b into its constituents. So you could have broken this up into 0 is equal to b squared. And then you could have broken it up into plus 6b, minus 10b, minus 60. And then factor it by grouping. Group these first two terms. Group these second two terms. Just going to add them together. The first one you could factor out a b. So you have b times b, plus 6. The second one you can factor out a negative 10. So minus 10 times b, plus 6. All that's equal to 0. And now you can factor out a b plus 6. So if you factor out a b plus 6 here, you get 0 is equal to b minus 10, times b, plus 6. We're literally just factoring out this out of the expression. You're just left with a b minus 10. You get the same thing that we did in one step over here. Whatever works for you. But either way, the solutions are either b is equal to negative 6, or b is equal to 10. And we have to be careful here. Remember, this is a word problem. We can't just state, oh b could be negative 6 or b could be 10. We have to think about whether this makes sense in the context of the actual problem. We're talking about lengths of triangles, or lengths of the sides of triangles. We can't have a negative length. So because of that, the base of a triangle can't have length of negative 6. So we can cross that out. So we actually only have one solution here. Almost made a careless mistake. Forgot that we were dealing with the word problem. The only possible base is 10. And let's see, they say find the height and the base. Once again, done. So the base we're saying is 10. The height is four inches less. It's b minus 4. So the height is 6. And then you can verify. The area is 6 times 10 times one half, which is 30.