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Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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30 Jun 2018, 11:10
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Question2 Which of the following equation has \(3+2\sqrt{3}\) as one of its roots? A. \(x^2+6x+3 = 0\) B. \(x^26x+3 = 0\) C. \(x^2+6x3 = 0\) D. \(x^26x3 = 0\) E. \(x^24 \sqrt{3}x3 = 0\)
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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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04 Jul 2018, 03:45
Solution Given:• One of the roots of a quadratic equation is \(3 + 2\sqrt{3}\) To find:• The original equation, with the given root \(3 + 2\sqrt{3}\) Approach and Working: We know that if one of the roots of a quadratic equation is \(a + \sqrt{b}\), then as a conjugate root, the other one will be \(a – \sqrt{b}\) Therefore, the roots of the given equation will be: • \(3 + 2\sqrt{3}\) • \(3 – 2\sqrt{3}\) So, the equation can be: • \([x – (3 + 2\sqrt{3})] [x – (3 – 2\sqrt{3})] = 0\) Or, \((x – 3 – 2\sqrt{3}) (x – 3 + 2\sqrt{3}) = 0\) Or, \(x^2 – 3x + 2\sqrt{3}x – 3x + 9 – 6\sqrt{3} – 2\sqrt{3}x + 6\sqrt{3} – 12 = 0\) Or, \(x^2 – 6x – 3 = 0\) Hence, the correct answer is option D. Answer: D
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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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30 Jun 2018, 19:54
Here's my solution: If one root is 3+2\(\sqrt{3}\), then other root is 32\(\sqrt{3}\). Sum of roots = 6 Product of roots = 9(4*3) = 3 Hence equation is: \(x^{2}\)  (sum of roots)*x + (product of roots) = \(x^{2}\) 6x 3 Hence D
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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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Updated on: 01 Jul 2018, 06:25
One of the ways is to approximate the value of \(3+2\sqrt{3}\) =3+2*1.7=6.4 Substitute 6.4 for x
Only option D. \(x^26x3 = 0\) will work with the value. Answer D.
Originally posted by Hero8888 on 30 Jun 2018, 15:04.
Last edited by Hero8888 on 01 Jul 2018, 06:25, edited 2 times in total.



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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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30 Jun 2018, 16:14
(X(3+2sqrt(3))(x(32sqrt(3)) =x^2(32sqrt(3))x(3+2sqrt(3))x3 =x^26x3 Answer D
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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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04 Jul 2018, 03:57
Alternate Solution Given:• One of the roots of a quadratic equation is \(3 + 2\sqrt{3}\) To find:• The original equation, with the given root \(3 + 2\sqrt{3}\) Approach and Working: • The value of \((3 + 2\sqrt{3})^2 = 9 + 12 + 12\sqrt{3} = 21 + 12\sqrt{3}\) Replacing the value of \(x^2\) and \(x\) in the given options, we get: • \(21 + 12\sqrt{3} + 18 + 12\sqrt{3} + 3\)
• \(21 + 12\sqrt{3} – 18 – 12\sqrt{3} + 3\)
• \(21 + 12\sqrt{3} + 18 + 12\sqrt{3} – 3\)
• \(21 + 12\sqrt{3} – 18 – 12\sqrt{3} – 3\)
• \(21 + 12\sqrt{3} – 24 – 12\sqrt{3} – 3\)
Hence, the correct answer is option D. Answer: D
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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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04 Jul 2018, 20:25
We are given that one of the root is 3+2√3, so another root is 32√3. (We know that if one of the roots of a quadratic equation is a+√b, then as a conjugate root, the other one will be a–√b) All the quadratic equation can be expressed as aX^2+bX+C=y Let the two different roots be X1 and X2 X1=3+2√3 X2=32√3 X1 * X2 =3 X1+X2 = 6
We also know that: X1*X2 = c/a X1+X2 = b/a
A. x^2+6x+3=0 X1*X2 = 3 X1+X2 = 6 Not the answer we are looking for. B. x^2−6x+3=0 X1*X2 = 3 X1+X2 = 6 Not the answer we are looking for. C. x^2+6x−3=0 X1*X2 = 3 X1+X2 = 6 Not the answer we are looking for. D. x^2−6x−3=0 X1*X2 = 3 X1+X2 = 6 Bingo, this is the answer choice we are looking for. E. x^2−43√x−3=0



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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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12 Mar 2019, 17:35
EgmatQuantExpert wrote: Solution Given:• One of the roots of a quadratic equation is \(3 + 2\sqrt{3}\) To find:• The original equation, with the given root \(3 + 2\sqrt{3}\) Approach and Working: We know that if one of the roots of a quadratic equation is \(a + \sqrt{b}\), then as a conjugate root, the other one will be \(a – \sqrt{b}\) Therefore, the roots of the given equation will be: • \(3 + 2\sqrt{3}\) • \(3 – 2\sqrt{3}\) So, the equation can be: • \([x – (3 + 2\sqrt{3})] [x – (3 – 2\sqrt{3})] = 0\) Or, \((x – 3 – 2\sqrt{3}) (x – 3 + 2\sqrt{3}) = 0\) Or, \(x^2 – 3x + 2\sqrt{3}x – 3x + 9 – 6\sqrt{3} – 2\sqrt{3}x + 6\sqrt{3} – 12 = 0\) Or, \(x^2 – 6x – 3 = 0\) Hence, the correct answer is option D. Answer: DCould you please explain why the other root must necessarily. be \(3 – 2\sqrt{3}\) ? I could theoretically have any other root and draw a parabola/quadratic that crosses both points on the x axis... e.g. y = (x  (\(3 + 2\sqrt{3}\))) * (x(\(7 + \sqrt{3}\))) ... In other words, I could tell you that one of the two roots in a quadratic is {insert any number} How does that fact alone tell you anything about what the value of the second root is? Wouldn't that entail some assumption about the shape of the parabola and where the point of reflection is?



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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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13 Mar 2019, 02:15
Hey levloans, We know that both the roots of a quadratic equation can be found out by applying \(\frac{b + \sqrt{b^2 4ac}}{2a}\). So, if one root is of the form \(\frac{b + \sqrt{b^2 4ac}}{2a}\) then other root will be of the form \(\frac{b  \sqrt{b^2 4ac}}{2a}\). Notice the difference in sign. Therefore, if one root is \(3 2\sqrt{3}\) then another root will be \(3+ 2\sqrt{3}\) . I hope this answers your query. Regards, Ashutosh
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Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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13 Mar 2019, 05:30
EgmatQuantExpert wrote: Hey levloans, We know that both the roots of a quadratic equation can be found out by applying \(\frac{b + \sqrt{b^2 4ac}}{2a}\). So, if one root is of the form \(\frac{b + \sqrt{b^2 4ac}}{2a}\) then other root will be of the form \(\frac{b  \sqrt{b^2 4ac}}{2a}\). Notice the difference in sign. Therefore, if one root is \(3 2\sqrt{3}\) then another root will be \(3+ 2\sqrt{3}\) . I hope this answers your query. Regards, Ashutosh Thanks for your quick response, Ashutosh. I had considered this pattern, but still do not see why it is necessary that our roots are these. Let me rephrase: I understand that \(3 + 2\sqrt{3}\) & \(3 2\sqrt{3}\) COULD be our roots. But what says that they HAVE to? i.e. For example Couldn't I have a quadratic with the roots (1) \(3+ 2\sqrt{3}\) and (2) 11 ? This is a real quadratic with xintercepts at \(3+ 2\sqrt{3}\) and 11... in fact, just look at this graph for visual reference https://www.wolframalpha.com/input/?i=y%3D(x(3%2B2*sqrt(3)))*(x11)Just because I know the first root (1), it doesn't seem to me that I can make assumptions about what the second root (2) is, as that could be anything. Or, I can know one of the x intercepts, but that doesn't tell me anything about what the other is, unless I know something else about the form of the quadratic. ... I can "peg" one of the xintercepts (roots) and then move the other one to wherever I want and still have a valid parabola/quadratic. I think your answer is in some way is affirming the consequent; or pulling in info from the output to make a judgment on the input, but the output is not given information, we can only start with the input of the question. Please let me know me if I am missing something.



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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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28 Jul 2019, 06:23
EgmatQuantExpert wrote: Question2 Which of the following equation has \(3+2\sqrt{3}\) as one of its roots? A. \(x^2+6x+3 = 0\) B. \(x^26x+3 = 0\) C. \(x^2+6x3 = 0\) D. \(x^26x3 = 0\) E. \(x^24 \sqrt{3}x3 = 0\) I think I have an easier solution than the ones posted here. x = \(3+2\sqrt{3}\) This can be rewritten as \(x  3 = 2\sqrt{3}\) Squaring both the sides, we get \(x^2+96x = 12\) Subtracting both sides by 12, we have \(x^26x3\), and hence, (D) is the correct answer choice PLEASE HIT KUDOS IF YOU LIKE MY SOLUTION



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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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28 Jul 2019, 07:11
EgmatQuantExpert wrote: Alternate Solution Given:• One of the roots of a quadratic equation is \(3 + 2\sqrt{3}\) To find:• The original equation, with the given root \(3 + 2\sqrt{3}\) Approach and Working: • The value of \((3 + 2\sqrt{3})^2 = 9 + 12 + 12\sqrt{3} = 21 + 12\sqrt{3}\) Replacing the value of \(x^2\) and \(x\) in the given options, we get: • \(21 + 12\sqrt{3} + 18 + 12\sqrt{3} + 3\)
• \(21 + 12\sqrt{3} – 18 – 12\sqrt{3} + 3\)
• \(21 + 12\sqrt{3} + 18 + 12\sqrt{3} – 3\)
• \(21 + 12\sqrt{3} – 18 – 12\sqrt{3} – 3\)
• \(21 + 12\sqrt{3} – 24 – 12\sqrt{3} – 3\)
Hence, the correct answer is option D. Answer: DI also did the above mentioned approach, however to reduce time on calculation, I did a bit of prethinking. While eliminating answer choices, it is important to cancel out the roots. Hence, while solving I require one positive root and the other negative root. This way, you'll be able to shortlist B & D. On plugging in value in D, you'll get the answer. Hence, via pre thinking, you didn't even have to solve A & C. Thanks
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Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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31 Jul 2019, 08:57
EgmatQuantExpert wrote: Question2 Which of the following equation has \(3+2\sqrt{3}\) as one of its roots? A. \(x^2+6x+3 = 0\) B. \(x^26x+3 = 0\) C. \(x^2+6x3 = 0\) D. \(x^26x3 = 0\) E. \(x^24 \sqrt{3}x3 = 0\) Which of the following equation has \(3+2\sqrt{3}\) as one of its roots? If \(3+2\sqrt{3}\) is one root, other root is \(32\sqrt{3}\) Sum of roots = 6 Product of roots = \((3+2\sqrt{3})(32\sqrt{3}) = 9 12 =3\) Equation \(x^2  6x 3 =0\) IMO D
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Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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31 Jul 2019, 09:31
nkin wrote: Here's my solution: If one root is 3+2\(\sqrt{3}\), then other root is 32\(\sqrt{3}\).
Sum of roots = 6 Product of roots = 9(4*3) = 3
Hence equation is: \(x^{2}\)  (sum of roots)*x + (product of roots) = \(x^{2}\) 6x 3
Hence D If the roots of x^2 + ax + b = 0 are x1 and x2, then: x1 + x2 =  b/a x1*x2 = c/a If x1 = 3+2√3, then x2 must be the conjugate of x1, that is 32√3 x1 = 3+2√3 x2 = 32√3 x1+x2= 6= b/a. a=1, then b=6. x1*x2= 3^2  (2√3)^2 = 912= 3. a=1 then c=3. Answer is obviously (D) x^2 + ax + b = x^2  6x  3 Posted from my mobile device




Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots
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