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Solving a Quadratic by Completing the Square - Concept
Solving quadratics can be difficult. Completing the square is one method for solving a quadratic equation. A quadratic equation can also be solved by factoring, using the square roots or quadratic formula. Solving quadratic equations by completing the square will always work when solving quadratic equations and is a good tool to have in your math tool belt.
Solving a quadratic equation by completing the square. So we already know how to use these square root properties to solve a quadratic. So if we're ever given with an equation with something already squared, all we want to do is isolate that term, so in this case x-3 squared is equal to 5. Take the square root of both sides. So we end up with x-3 is equal to plus or minus root 5. Remember whenever you use the square root as a tool we have to include plus or minus. Then solve for x we just to add 3 to both sides leaving us with x is equal to 3 plus or minus root 5, okay.
So the square root property is a really handy property when we have something squared, okay? The problem is is that we don't always have something squared, okay? So we're going to go to another example where we are going to use this completing the square in order to get it in this form. Okay.
So what we want too do is to turn this problem into something squared, okay? The first step that we want to do is isolate all our x terms together. So what we want to do then is subtract the 10 over x squared plus 8x is equal to -10. Okay. So I now want to turn this piece into something squared, okay? The x squared and the 8x are fixed. I can't change those. Okay? I left a little space at the end because we can add something to both sides and our problem doesn't change.
So what we want to do is figure out what we can add there in order to make a perfect square including this 8x, okay. So I know that this has to be a x and it has to be a plus. Okay? This middle term is positive so that tells us it has to be a positive sign. But what we want to do is somehow figure out what we can put here in order to get 8 and our middle term if we foiled it out, okay? And the trick for that is you take this middle term and divide it by 2. Okay? So in this case 8 divided by 2 is 4. That is what is going to go right here. Okay? And what happens when we foil this out what we end up getting is x squared plus 8x plus 16.
So what I've really done is by including this 4 in here I've added 16 to my initial equation. So I've added 16 to this side, I also have to add 16 to this side to keep it balanced. Whatever we do to one side we also have to do to the other. Okay? So what we actually have in this case then is x+4 quantity squared is equal to 6. Okay?
So this is what's called completing the squares, okay. We took our term, figured out what we needed to add to both sides to make it a perfect square. So that middle term divided by 2, that goes in here and then that new term squared is get what gets added in both sides and then we're able to rewrite this as a perfect square.
Once we get to this point we use the same exact method we did at the very beginning in order to solve this. Take the square root of both sides x+4 is equal to plus or minus the square root of 6. And then solve for x in this case subtract 4 so we end up with x is equal to -4 plus or minus the square root of 6. So I completed the square what we're able to do take is take a quadratic equation that couldn't be solved by using the square root property, turn it into a perfect square so that you could use the square root property in order to solve it.