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 + 6x = 7

Accepted Answer

Welcome back. I'm so glad you're here. We have this equation X squared plus 12 X equals 13. And we are to solve it by completing the square while typically when we're solving by completing the square, we look for the format A X squared plus bx plus C equals zero. Where A corresponds to the number in front of X squared B corresponds to the number in front of X and C is our constant term. This is very similar to that, except our constant term is on the right hand side and it's not equal to zero. We can still work with this. The next question we ask ourselves is does a equal one? Just the number in front of X squared equal one. And yes, in this case it does. Well that's great. This makes for a lot fewer steps. The next step we would take because A equals one is we would move our constant term over to the right hand side of the equation. And look that's what they already did for us. So that's great. Now we're left with X squared because A equals one plus B X. And then we leave a space where the sea term was equals a negative C. And we leave a space after that C term. So let's rewrite our equation leaving that space. So we've got X squared plus X. Space equals 13. Space. What do we fill in those spaces with what we're going to do because A equals one. We're going to take B divided by two and square it. So what's our B term again? That is 12. We're going to divide 12 by two And square it. What's 12 divided by two? That is six. And before we do the squaring, I want you to pause when you've got a number simplified inside the parentheses, pause and take note of that number. And then once you've taken note, you can move on because we do need to square it. So six squared is 36. That is the number that goes in the space. So let's rewrite it with the number in the space. We have X squared plus 12. X plus 36 equals 13 plus 36. Be sure to add that to both sides. Well, great, X squared plus 12 X plus 36 we can factor that and 13 plus 36. Well that's 49 on the right hand side of the equation. What times what gives us X squared. That's X times X. And what? Two factors when multiplied equal, positive 36. And when added equal to positive 12, that's plus six plus six. Or in other words X plus six times X plus six is X plus six squared equals 49. Look at that perfect square And hey look at that number inside the parentheses. That's the same as the number we took note of Not too long ago. That's because when we're doing um completing the square and when A is equal to one. When you have X squared plus Bx plus B over two squared. That equals parentheses, X plus B over two and parentheses squared. There's that number of which we took note. Great. Well now let's go ahead and solve for X on the left. So we've got X plus six squared we'll take the square to that. What we do to one side we do to the other. So the squared of X plus six squared is just X plus six and that equals when we took the square root. So we do a plus or minus. The square root of 49 is seven. Now to get that X by itself, we'll just subtract six from both sides. Note that we have two equations here, we have X equals the positive seven minus six and we have X equals the negative seven minus six from that plus or minus. Well seven minus six, that's one. So X equals one and negative seven minus six is negative 13. So X also equals negative 13. We look at our answer choices and that matches with answer choice A Well done, we'll catch you on the next one

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