 This video is called Solving Proportions 2. What we're trying to figure out in a problem like this is what does x have to be so the fraction on the left equals the fraction on the right or so they're proportional or equivalent to each other. So remember, if you ever see two fractions with an equal sign between them, that's what you're trying to do. Figure out how those fractions can be equivalent. So always start with your cross multiplication. So it looks like we're going to have to do x minus three times the quantity of x minus seven. I put parentheses around those simply to make sure I don't forget about anything and I remember the x minus three came from one side, the x minus seven came from the other fraction and then that equals 12 times 8 which is 96. So now it looks like my next step to figure out what x is, I'm going to have to clean up that left hand side so I'll have to foil. First, outside, inside, last. So x times x gives me x squared, x times negative seven is a negative seven x, negative three times x is a negative three x and then when I multiply the last, negative three times negative seven is a positive 21. That whole thing equals 96. I'll very quickly clean up that left side a little more by combining like terms x squared minus 10x plus 21 equals 96. Now where do I go from here? How do I solve for x? Lots of students get stuck at this part and hopefully you can recognize that when you see the exponent of two that you know you have a parabola and that to solve this you're going to have to factor and in order to factor you need everything on the same side of your equation. So we're going to move the 96 over so I'll subtract 96 from both sides so I end up with x squared minus 10x minus 75 equals zero. So now to continue our factoring we're trying to figure out what multiplies to give us a negative 75 and adds to give us a negative 10. I can go ahead and get started. I'm going to split up the x squared to x and x and now I'm going to make myself a little cheat sheet. What are some factors of 75? Well we could do 75 and 1, 15 and 5, 25 and 3 are the ones I can think of. Would any of those multiply to a negative 75 and add to a negative 10? I think they do. If we had a negative 15 and a positive 5 that's going to add to a negative 10 and a negative 15 times 5 multiplies to negative 75. So I found the correct factors by using a negative 15 and a positive 5. But now I'm not quite done yet because remember I'm still trying to solve for x. I have that equal zero to take care of. So I split up my problem into two, x minus 15 equals zero and x plus 5 equals zero. So when I add 15 to both sides I get x equals 15 and when I subtract 5 from both sides I get x equals negative 5. So this problem I had two answers. We could confirm that by plugging it in back at the beginning. I'll give myself a little bit of room here. If you didn't have that written down yet just pause your video and rewind. So let's see. If x is 15 we'd have 15 minus 3 over 8 equals 12 over 15 minus 7. Well 15 minus 3 is 12 so we have 12 over 8 and 15 minus 7 is also 8. So we ended up with 12 over 8 equals 12 over 8. So we just confirmed that x equals 15 works as an answer. Let's try the same with our negative 5. I'm simply plugging negative 5 into every time I see an x from my beginning problem. Let's see what happens. Negative 5 minus 3 is a negative 8. So you end up with negative 8 over 8 which equals a negative 1. And over here negative 5 plus a negative 7 is a negative 12. 12 divided by negative 12 is also a negative 1. So you can see that both of those fractions ended up equaling the same thing. So if x equals negative 5 we've also solved the proportion. So here is an example of a problem that has two answers.