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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.$$\frac{1}{2} x^{2}-\frac{3}{4} x=\frac{5}{6}$$

$$\frac{9 \pm \sqrt{321}}{12}-0.743,2.243$$

Algebra

Chapter 0

Reviewing the Basics

Section 3

Completing the Square

Equations and Inequalities

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Solve each of the quadrati…

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In each of the following, …

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When we look at this quadratic, what we need to do is get, uh and we're completing the square. So we need that leading coefficient to equal one is equal to 56 So if I want this leading coefficient equal one, I can just multiply everything by two, which is perfectly acceptable. Um, and then I would simplify this, like two goes into four twice sort of looking at three halves. I'm gonna put a blank right here and again to goes into 63 times. So on the right side, I have equals five thirds. But again, I'm putting a blank because I want to complete the square, which is taking this be value. And, uh, instead of dividing by two of them multiplied by one half. So you take half of the b value squared, um, so that's equal to negative three force. When I square that well, negative squared is positive. Three squared is 94 square to 16. The only way to do that is to add the same fraction to the other side. And so now what we have is a perfect square. Try no meal on the left side and the shortcut. It's always this number. So it's X minus three. Force that quantity squared, and on the right side, you just might need to get the same fraction. So looking at five thirds plus 9 16, which is 107 Yeah, 48 slips Looks kind of bad. Mhm. So now I can square root both sides and as I square root both sides. Um, well, the square root in the square Cancel. I would think of this denominator as Route 16 times, Route three, because I know the score to 16. It's four, but then I can rationalize the denominator. Multiply top and bottom by Route three. So on the left side, I just have X minus three force and on the right side have to figure out what 100 and seven times three is. It's a root. Don't forget about plus or minus 321 and in the bottom I have square to 16. Us. Forward in the square root of nine is three or four times. We would give me 12, so it might be to your benefit to get this denominator to B 12 as well, so you can just add them together, um, side to multiply four by three to get to 12. So three times three is nine. So, as I add that to the right side, I get an answer of nine plus or minus Route 3. 21 all over 12. I do not think that is simplify about now. This is an irrational number. So we can practice doing nine minus Route 21 all over 12 to get a definite answer. Uh, which is negative? Point 743 And then I can also do nine plus route 3. 21 all over 12. 243 That's just using a calculator to get that answer. There you go.

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