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Solving a Quadratic by Completing the Square - Problem 1
Solving a quadratic equation by completing the square. What we’re going to do now is solve the equation behind me by completing the square and the first thing we always want to do is doing this is to isolate our x terms. So we want to get every other constant term to other side, leaving just our x terms. X² minus 4x is equal to 32.
We now want to figure out if there’s a way to write these two terms as something squared. So we want to say this is going to be x, this is minus so I know this is a minus sign in between and I want to somehow figure out how to express this as a square. The way we do that is we take our middle term and divide it by 2. So 4/2 is 2, that is what’s going to go here. That new term we just wrote in if going to be squared and that’s going to be added inside of that. The way you can test this out, if we FOIL this out we end up getting x² minus 4x plus 4.
What happened is that in order to express this part as a perfect square I had to add 4 to one side, to keep a balance I have to add 4 to the other side as well. What we’ve ended up with is x minus 2 quantity squared is equal to 36.
We now are able to use the square root property. We have something squared is equal to a number, take the square root of both sides, leaving us with x minus 2 is equal to plus or minus 6. Add 2 to the other side and what we end up with is if we have a +6 plus 2 we get 8. If we have a -6 plus 2 we get -4. So by completing the square we were able to solve out this equation.
There’s I want to bring your attention to before we do that. This problem actually we could have factored just from the get go. We come over here, whenever we have a quadratic equal to zero, if we can factor it we always should it's a little bit easier than going through this process, and this is just a x minus 8 times x plus 4 is equal to zero giving us the answer of 8 and -4 which is what we found over here.
So two different ways of solving the exact same equation obviously factoring in this case is a little bit easier but the process of completing the square is really important and is going to come up later in your class as well.