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Solve each of the quadratics by first completing the square. When the roots are irrational, also give the solutions to the nearest one thousandth.$$x^{2}-6 x+23=0$$

$$3+\sqrt{14 i}$$

Algebra

Chapter 0

Reviewing the Basics

Section 3

Completing the Square

Equations and Inequalities

Campbell University

McMaster University

Idaho State University

Lectures

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Solve each of the quadrati…

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In each of the following, …

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So we're looking at X squared minus six X plus 23 is equal to zero. And whenever you're completing the square, I just think it's a good habit to put a blank on each side of your equation and to get rid of the constant term. So I'm going to subtract 23 to the right side. Yes, I kind of skipped a step because zero minus 23 is negative. 23. And now we're ready to complete the square by taking that be valued negative six divided by two and then square. That number may have six. Divided by two is negative. Three. When I square that number, I'll get positive night. So why do we go through? That process is now The left side is a perfect square. Try no meal factors of nine that had to be negative. Six. Well, it's always this number. Negative three. Negative three. And instead of writing X minus three twice, we can write it Once squared on the right side, we can actually figure out the negative 23 plus nine. Give me negative 14. Now here's the thing is a and if we're going to square a number we have to get a positive. So that's my clue that we're going to have a non real solution, so I'm not imaginary complex. Um, And so now when we square root both sides the square root in the square root council and we're left with X minus three on the right side, we would just have plus or minus I route 14. So, uh, now we can't we I guess we could break down 14, but there's no perfect squares in there two times seven. And neither of those are perfect square. Uh, those are all the prime numbers, so we're done breaking that down. So there's only one thing left to do, and that's to move this real number of negative three to the other side. And you do that by adding three over. And we always write that real part in front of the imagination. And this is how I like to write my answer. Um, it's complex, so there's no way of writing the irrational solutions. This is it

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