Solving Rational Equations with Like Denominators - Problem 3

Here I have two equal fractions, so a lot of students will look at this and think about cross multiplying. Remember guys a short cut you can use though when you have the same denominators is to multiply both sides by that denominator. Like if I multiply by x minus 7 and by x minus 7, it’s essentially like getting rid of my fractions that makes me a happy camper.

If I can use that trick, then this is just going to be a straight forward solving problem. This is going to be a quadratic equation because I have an x² term and you guys know lots of different strategies for solving quadratics. You might choose to graph, take square roots of both sides, but that’s not a good idea because I have a b term. You might factor a quadratic equation or complete the square. Any of those are options for you.

I’m pretty good at factoring, that’s my favorite method so I’m going to try that especially because my leading coefficient is 1. If I had a different coefficient right there, factoring might not be so easy. Well let’s check it out.

I need numbers that multiply to -3 and add up to -2 so that’s going to be -3 and +1. Good that was a good factoring problem. If I couldn’t factor, I would probably use the quadratic formula. Okay so what I’m going to do is use the zero products property and set each one of those factors equal to 0 to find my answers. X is equal to 3 and x is equal to -1. Before I move on I want to make sure these are not excluded values.

It would be an excluded value I it would make my denominator equal to zero, but neither one of these makes my denominator equal to 0, those are my two solutions. I could check these by going back and substituting one at a time. X equals -1, x equals 3 and making sure that I do in fact get equivalent fractions. I’ll leave that to you guys to check on your own.