Solve each equation in Exercises 47–64 by completing the square.
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Question

Solve each equation in Exercises 47–64 by completing the square.

x^2 - 2x = 2

Accepted Answer

Welcome back. I'm so glad you're here. We have the equation X squared minus X equals seven more to solve for X by completing the square. Alright, typically when we need to complete the square, we will be looking for something in the format of a X squared plus bx plus C equals zero. Where A corresponds to the term in front of X squared B corresponds to the term in front of X and C corresponds to the constant term. So this is very similar, but in a slightly different format than X squared plus bx plus C equals zero because this one doesn't equal zero. Well, the next step we would be taking and completing the square is asking ourselves does a equal one and in this case yes A does equal one. The number in front of X squared is one. So great. We don't have to worry about doing anything with the A. And the next thing we would be doing is actually moving the constant term, R C term over to the right hand side of the equation. And look that's what they did here for us. So that step is saved for us. We would normally then have X squared plus B. X. And then we would leave some space and equals negative C. And then a little bit more space. So let's rewrite what we've been given with that extra space in there. So it'll be X squared minus 20 X. Leave some space equals seven. Leave some space because the next step in completing the square is to fill in that space. What do we fill it in with? Because a equals one, we're going to take B divided by two and square it. So our B term here is negative 20. We'll divide that by two And square it negative divided by two. That's negative 10. And before you square it, when you've got a simplified number inside the parentheses. But before you've squared it, I want you to pause and take note of what that number is And once you've taken note you can move on because we do need to square it. So negative 10 square that is 100 and that is the term that is going to get filled into the space. So let's do that. We now have X squared minus 20. X plus 100 equals seven plus 100. And look at that X squared minus 20 X plus 100. We can factor that. And on the right hand side, seven plus 100 that's 107. Well, what times what gives us X squared? That's x times X. What? Two factors when multiplied? Give us a positive 100. But when added give us a negative 20 that's going to be minus 10 minus 10. Or in other words X minus 10 times X minus 10 is x minus 10 squared. Look at that perfect square And hang on a second. Look at that number inside our parentheses, That's familiar. That's the number we took note of not too long before. And that's because when we have when a equals one, when we take X squared plus bx plus B over two squared. By definition, we've got this parentheses, X plus B over two and parentheses squared. There is that number of which we took note. Great. Well now let's solve for X. We can take the square root of both sides because the square root of x minus 10 squared is x minus 10. And then on the right hand side that will be a plus or minus. And simplified the square root of 107 is the square root of 107. Now to get X by itself will just add 10 to both sides. And take note that there are two equations here because one is X equals positive square root of 107 plus And one is x equals negative square root of 107 plus 10. And those two are not like terms, they're not going to combine but we can rewrite them so that it matches with our answer choices. And then we would have x equals 10 plus the square root of 107 And we have x equals 10 - the square root of 107. We look at our answer choices and that matches with answer choice. A Well done. We'll catch you on the next one

Solve each equation in Exercises 47–64 by completing the square. x^2 - 2x = 2

Key Concepts

Completing the Square

Quadratic Equations

Perfect Square Trinomial

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