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The quadratic equation whose roots are the minimum value of $sin ^{2} heta-sin heta+frac{1}{2}$ and $lim _{x ightarrow infty} sqrt{(x+1)(x+2)}-x$ is(a) $3 x^{2}-7 x...

Question

The quadratic equation whose roots are the minimum value of $sin ^{2} heta-sin heta+frac{1}{2}$ and $lim _{x ightarrow infty} sqrt{(x+1)(x+2)}-x$ is(a) $3 x^{2}-7 x+3=0$(b) $8 x^{2}-14 x+3=0$(c) $x^{2}-7 x+3=0$(d) $2 x^{2}-7 x+3=0$

The quadratic equation whose roots are the minimum value of $sin ^{2} heta-sin heta+frac{1}{2}$ and $lim _{x ightarrow infty} sqrt{(x+1)(x+2)}-x$ is (a) $3 x^{2}-7 x+3=0$ (b) $8 x^{2}-14 x+3=0$ (c) $x^{2}-7 x+3=0$ (d) $2 x^{2}-7 x+3=0$



Answers

For what values of $ a $ and $ b $ is the following equation true?
$$ \displaystyle \lim_{x\to 0} \left( \frac{\sin 2^x}{x^3} + a + \frac{b}{x^2} \right) = 0 $$

Even limit off. Now Sign do acts of a expert tree because I just being over x square An excuse. Juzo, Won't you find that I am? Be just Then we want to have this limit Echo Jews there are here and that you do this one first. I will want you would the limit here into one super fraction. So we the common denominator it could expert three. They will have the sigh your angst. No, I think about three but three x ondas execute you zero We have the farmers over zero if we're gonna Bender Lupton rule now! And we had a limit under express jeans. Oh, then by the Leviton relation an echo Judah Uh, close. I'd your ex on better general religion attempts to kill a front I know. Plus three a square, uh, three x squared. Blessed be over the tree Thanks square and everybody that x equal a chooser to this one Begin the form of the, uh this one will be, Uh ah! This one movie in it you get this will be equal to zero way required that the time He must also be coaches Aarhus Well so doesn't want to have the two times go, Sanders or it could you want Suciu? Plus no, B must echo. Juzo. That frumpy must be equal to minus two and now isn't implies. Then we should get now Limit on the X Guest Ju zero. And then we have the two course I off the Jew. Explicit tree and X square minus two. Now say value Military X square. And then, uh, now if it be, Your Majesty, we have the funders over zero. Therefore, we're gonna bladder live with and rule one more time. And then we could get by left in room. We have the limit on the express to zero. And now they lived on the top. You get negative minus for Cy for our site, your ex and then plus six X They writing by the six. Thanks. And here it extra good. Jizo. We required that. We have a zero in the denominator. Definitely required that This one I'm so need u p e coach is zero. Therefore, we have required that the man is far Say'i'm does equal to save zero last Ah six x could So So it also true here and then we should have the phone is there are Mazar here and by the loved in room began the limit Exodus, Juzo! And now the riv them the cycle Jew The because I said we got a minus eight because I under far and then plus six I divided by six And here we want this limit ico Juzo even Only if the numerator here must be coaches when executives there are even if the minus paid. So it's time zero echoed you one plus six I must equal zero Doesn't implies that that I must ico to uh yeah. Eight number six and equal to the four out of three. So we found out that I equal to find out of three and the three most echo two months to hear.

Okay, So to solve this quadratic equation, So I want to start by saying let X be equal to sign data, so sign data. Okay, so I'm gonna replace all the signs with X. So my new creation becomes three X squared, minus seven. Hex minus three is equal to zero. So naff we apply the quadratic formula. So quadratic formula says, um, X is equal to negative B. In this case, B is negative. Seven. So negative business with positive seven is, uh, negative or positive. Seven plus or minus the square root of B squared. So seven negative seven square, which is 49 minus four times a, which is three times See, which is negative. Three and then all over two times A, which is three. Okay, so now if we just try toe, um, simplify this. The answer. So this is equal to seven. Poster minus. Okay, so here we have the squared of 49 minus four times three times now, So this year's not be positive. So four times through just 12 times three is 36. So we have 49 plus 36 and then this is all over two titles. Three which is six. Could we superfly this one more time? So we get seven plus or minus the square root of so 49 plus 36 which gives us 85. No, this is all over since. Okay, so now remember, this is X. So from appear accessible to sign data, so X is equal to sign data, which is also equal to this. Um, this enter that we got here. So seven plus and minus squared of 85 over six. Okay, so now we want to try and solve member solving data. So the data to solve data here, we're gonna take inverse sign of this right here. So seven plus or minus the square of 85. What? We're six. Okay, so now if we punch that in our calculator, we'll get that data, um, fatal will be able to 3 38 mrs and degrees. And so 3 38 degrees is also negative. 22 degrees. But if I take the positive, um, 22 degrees and add 1 80 I also get to go to where you could do that just by doing the seven minus the square root of 85/6. Some of this will be the two. The answers that we get for a Fada. And you can write these, um, these angles in radiance. You just multiply by pi over 1 80 But if you want and degrees are a result will be 3 38 degrees at 202

Were provided with this Trigon Amit function and were asked to use the quadratic formula to solve with data. And they want data between zero and 360 degrees. So what? We're going to start with us? We know that a is equal to eight. B is equal to a negative seven and C is equal to one. So our quadratic formula is minus B. So that'll be seven plus or minus the square root of B squared. So that'll be 49 minus four times eight and then time See? So this one divided by two times a. So here we get seven plus or minus. The square root of 49 minus is 32 right? So that gets us 17. And that's over 16. And so are two solutions. Here are seven. So will be seven plus the square root of 17 over 16 which is equal to 0.69 519 and then we'll have seven minus the square root of 17 over 16 which gives us a positive number as well. And so both of these fall between negative one and a positive one, which is a necessity for co sign. And so now we just take the ark side or co sign of both of these. So I get co sign negative to sign for two, uh, Saul for data. Right. Um, let's of this value here and the number that I get here this 46 degrees. So when we think about our unit circle, where co sign remember its coastline sign? So where Cho Sinus positive is where X is positive. So that's over here, because this is our X, and so this is in the first and our fourth quadrant. So this isn't a first quadrant here, so this is 46 degrees, So it's right about there. And so we're also gonna have the reference angle over here at 46 degrees. So we take 360 and subtract 40 sex, and we get 314 degrees. And then when I take this other value here, I end up getting 80 degrees. So are 80 degrees is close to I guess I could use a different color. Here are 80 degrees is up here, and then our other reference angle. That show makes coast. I'm positive is over in this fourth quadrant, and so What you get is you want to just take 360 subtract 80 and I get 200 and 80 degrees. So my solution set for data between zero and 360 is 46 80 to 80 and 314.

Were provided with this equation here and were asked to use the quadratic formula to solve for theta Worth data's between zero and 360 degrees. So here, when you have two fractions equal to each other, you can guess cross multiply. So we'll start here with science squared times, sweets and parentheses. Here, so remember to distribute. So I get sign squared data plus two. Signed beta is equal to six, cause I'll cross multiply these guys here and then if I subtract six from both sides, I get signed. Square data plus two signed data minus six equals zero. So that means that a is equal to one. He is equal to two and C is equal to a negative six. So using our quadratic formula week at minus B plus or minus the square root of B or B square. And sorry, this weird of B squared, which is four minus four times A, which is one time C, which is a negative sex. And this is all over two times A. So I get a negative to plus or minus. But this is the square root of 28 over to when I reduced this I get negative to plus or minus two square roots seven over to. So that's negative one when I reduced this plus or minus the square root of seven. So when you take negative one plus the square root of seven, we got a 1.645 And this cannot be a value because the sign oscillates between a negative one and a positive one. So this is outside that range. So if we take a negative one minus the square root of seven here, this is a negative 3.646 So this also has the same problem. So this equation here has no solution.


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