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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.$$9-3 x^{2}+5 x=8+2 x^{2}$$

$$\frac{5 \pm 3 \sqrt{5}}{10},-0.171,1.171$$

Algebra

Chapter 0

Reviewing the Basics

Section 3

Completing the Square

Equations and Inequalities

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:36

In each of the following, …

02:50

Solve each of the quadrati…

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02:20

02:12

02:25

02:41

02:42

02:36

04:01

02:44

02:06

02:01

02:13

03:15

02:19

So we have this quadratic nine minus three X squared plus five X, yeah. Equals eight plus two X squared. Now I like to have positive terms, So I would probably add this three X where, I guess a positive leading coefficient. I'm trying to say, um but it actually doesn't matter where you put things. So may I would just like five x squared over here. Uh, now, because I made this a positive term, I actually have to move that negative five x to the right side. Even though I'm writing it on the left side. I hope that makes perfect sense. It's like I'm subtracting things over here. Um, and now I'm moving this eight to the left side. But then I'm writing it on the right side. I know that seems weird, but that's that's what I'm doing things. Um, and maybe you would write it the other way first, like one and then equals canceled, canceled, canceled. Um, and you have five x squared minus five x and then use the symmetric property. I don't know what you would do, but what I would do next is your leading coalition needs to be a one so I would just factor out a five on the left side. And I'm not changing the value of anything. So it's still one on this side. I didn't not dividing everything back. So the next thing I would do is look at my B value, take half of it, and then square that number. Um, well, you square one, you get one or square negative one and positive one and square to you get four. So I'm gonna add 1/4 to the side. But what I have to do is think of the quantity that I'm adding here because I'm not. I'm writing 1/4 but I'm actually adding, if I wouldn't distribute back in here by force as a quantity to the left side. So why do I go through that process is then on the left side. I can see I have, um, a perfect square trying to, you know, here's your shortcut is always X minus one half being squared. It's always this number is your shortcut. And on the right side that I have really one is the same thing as four force plus five force on the right side. I really have nine force. But as far as simplifying those, my next step would be to divide both sides by two black by five. Well, this is actually the same thing. It's multiplying by 1/5. The reason why I do that sometimes to show people that it just gets stuck into the denominator X minus one half Being squared is equal to nine. I'll just leave this four camps five, because as I square root each piece. Um, all right, so probably to rationalize the denominator. Anyway, I'm looking at X minus one half is equal to plus or minus square to minus three. Square root of four is to I need to rationalize that Route five route five times have to multiply top and bottom on my Route five. And let's get the same denominator I want. This is going to be two times five. So that's going to be a denominator of 10 and 3 to 5 in the numerator. So on the left side, I'm gonna rewrite one half is 5/10. I like that a little bit better. Um, that way, when I add 5/10 to the other side of my denominator is already the same. So I just need to write plus or minus three. Route five right there. So this should be your correct answer. Otherwise, what you might be asked to do is to divide at five minus three or five. All over 10 to give you negative 0.170 Oops. 171 So four or five plus three or five all over town. There should be 1.717 So those are your two irrational answers. If you worked it out that way,

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