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In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.$$f(x)=x^{2}-6 x+6$$

a) graphb) The roots of $f$ ROUNDED TO THE NEAREST TENTH are the $x=1.3$ and $x=4.7$ as shown in the pic.c) $x=3+\sqrt{3}$ and $x=3-\sqrt{3}$

Algebra

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Section 1

Real Roots of a Quadratic Equation

Equations and Inequalities

Quadratic Functions

Complex Numbers

Polynomials

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

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case are right here we have the function Experiment six Experts six. Let's start by sketching the graph of this, and you'll notice that if you take a look at the standard form, you'll see that the coefficient of the terms of positive one. You know that the graph is there if we're gonna have to open up, so we'll start by freeing the Vertex. So the Vertex should be X equals negative B over to A, which is the second term times negative one, which is six over two times the first term, which is just a one 76 over to, and that's a positive three. Let's play that in to find the white term of the vortex and after three equals nine minus 18 plus six, which is another 32 This time it's negative, and we have what seems to be a 1 to 1 scale right here. So we graft that. We end up three. Make it two three. So let's start by finding some of the points on either side. Let's try a four so recalls. One is for right, so I'll be 16 minus 24 plus six. Just, um, negative eight plus six inches and negative, too. Um, two equals four minus 12 plus six. Another negative, too. So skipping two in there f of five equals 25 minus 30 plus six. So be one same thing on the other side. Except it's one this time one minus six negative five plus six, which should be just a one again. So one in either said. I think two points and other side's good enough structure because I don't have any more room on the right side of the graph. Cells just sketched this right now. We fixed that. So it goes through that first point right there. Student, other side as well. First, it struck the area. Now let's move to the other side. And now, if we estimate where this function is zero as private, fastest by the roots, we can do that. And if you'll notice that since we have 2040 60 80 in terms of the 10th 2468 in between each of the whole numbers on this on this scale, we can estimate so on the left side it will be around one 0.2 euro and on the right side, this will be a little harder because of the arrow, but it should be around a nine over there. So since seems to be in between E. T and 100 should should be a 34.9 your own that right side. So roots X equals 1.2 and 4.9. We make that look less like currencies, then part See, we actually have to find the roots. In order to do that, we have to find our radical form, and we have to move this into apologetic form. So we have to use our handed any quadratic formula, which states X equals negative B plus or minus the route B squared minus or a C all over two A. That means it's negative one times negative. Six. Just six. What's the minus? The roots 36 minus four times one times six minus 24 All over two. The chief was 6% minus 23 over. To which equals three plus or minus. Recruited me the vial those numbers to to simplify. And that is our answer

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