Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 47983

Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
15 Jun 2016, 01:40
Question Stats:
64% (01:21) correct 36% (01:22) wrong based on 1264 sessions
HideShow timer Statistics




Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2759

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
03 Sep 2016, 09:00
Bunuel wrote: Which of the following equations has 1 + √2 as one of its roots?
A) x^2 + 2x – 1 = 0 B) x^2 – 2x + 1 = 0 C) x^2 + 2x + 1 = 0 D) x^2 – 2x – 1 = 0 E) x^2 – x – 1= 0 To solve this problem, we need to use the following two facts: 1) If a quadratic equation has integer coefficients only, and if one of the roots is a + √b (where a and b are integers), then a  √b is also a root of the equation. 2) If r and s are roots of a quadratic equation, then the equation is of the form x^2 – (r +s)x + rs = 0. Since we know that 1  √2 is a root of the quadratic equation, we can let: r = 1 + √2 and s = 1  √2 Thus, r + s = (1 + √2) + (1  √2) = 2 and rs = (1 + √2)(1  √2) = 1 – 2 = 1. Therefore, the quadratic equation must be x^2 – 2x – 1 = 0. Answer: D
_________________
Jeffery Miller
Head of GMAT Instruction
GMAT Quant SelfStudy Course
500+ lessons 3000+ practice problems 800+ HD solutions




Intern
Joined: 03 Jun 2015
Posts: 3

Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
15 Jun 2016, 02:03
Bunuel wrote: Which of the following equations has 1 + √2 as one of its roots?
A) x^2 + 2x – 1 = 0 B) x^2 – 2x + 1 = 0 C) x^2 + 2x + 1 = 0 D) x^2 – 2x – 1 = 0 E) x^2 – x – 1= 0 If x1 = 1 + √2 then x2 = 1  √2. By the Viete theorem: b = (x1 + x2) = (1 + √2 + 1  √2) = 2 c = x1 * x2 = (1 + √2)(1  √2) = 1  2 = 1 is the answer.




Manager
Joined: 07 May 2015
Posts: 175
Location: India
GPA: 3

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
15 Jun 2016, 18:46
A and C are automatically out as they are roots of 1 and 1 respectively
for rest of option substitute 1  root(2)...which ever equation gets satisfied is the answer.
Only equation D satisfies
Ans: d



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8195
Location: Pune, India

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
15 Jun 2016, 23:23
Bunuel wrote: Which of the following equations has 1 + √2 as one of its roots?
A) x^2 + 2x – 1 = 0 B) x^2 – 2x + 1 = 0 C) x^2 + 2x + 1 = 0 D) x^2 – 2x – 1 = 0 E) x^2 – x – 1= 0 Note that (B) is (x  1)^2 so its roots are 1 and 1. (C) is (x + 1)^2 so its roots are 1 and 1. Now note that all 3 of the remaining options have x^2. When you put \(x = (1 + \sqrt{2})^2\) in them, you will get \((+2\sqrt{2})\) term. It should get canceled out by another term to get 0. So you should get \((2\sqrt{2})\) term. Option (D) has 2x which will give a term \(2\sqrt{2}\). So answer (D)
_________________
Karishma Veritas Prep GMAT Instructor
Save up to $1,000 on GMAT prep through 8/20! Learn more here >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Manager
Joined: 06 Jun 2014
Posts: 90
Location: United States
Concentration: Finance, General Management
GPA: 3.47

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
03 Jul 2016, 16:10
fla wrote: Bunuel wrote: Which of the following equations has 1 + √2 as one of its roots?
A) x^2 + 2x – 1 = 0 B) x^2 – 2x + 1 = 0 C) x^2 + 2x + 1 = 0 D) x^2 – 2x – 1 = 0 E) x^2 – x – 1= 0 If x1 = 1 + √2 then x2 = 1  √2. By the Viete theorem: b = (x1 + x2) = (1 + √2 + 1  √2) = 2 c = x1 * x2 = (1 + √2)(1  √2) = 1  2 = 1 is the answer. How can I solve option A, using Vieta's Theorem?
_________________
1) Kaplanprep 450 Q27 V21 2) Manhattan 530 Q35 V28 3) GmatPrep 450 Q33, V19 4) Veritas 460 Q31, V23 5) Veritas 440 Q 30, V21 6) Veritas 500 Q34, V 25 7) Gmat 420 Q27, V23 8) Veritas 520 Q36, V26 2/2 9) Veritas 540 Q37, V28 4/19 10)Manhattan 560 Q40, V28 4/28



Intern
Joined: 04 Jun 2016
Posts: 7

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
03 Jul 2016, 20:58
I like to think about this question in this way: As irrational roots occur in pairs, the other root has to be 1\sqrt{2}. So the unknown coefficients would be ()sum of roots and (+)product of roots. This gives us D. Is this a good approach for these kind of questions?



Intern
Joined: 03 Jun 2015
Posts: 3

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
04 Jul 2016, 01:05
zxcvbnmas wrote: fla wrote: Bunuel wrote: Which of the following equations has 1 + √2 as one of its roots?
A) x^2 + 2x – 1 = 0 B) x^2 – 2x + 1 = 0 C) x^2 + 2x + 1 = 0 D) x^2 – 2x – 1 = 0 E) x^2 – x – 1= 0 If x1 = 1 + √2 then x2 = 1  √2. By the Viete theorem: b = (x1 + x2) = (1 + √2 + 1  √2) = 2 c = x1 * x2 = (1 + √2)(1  √2) = 1  2 = 1 is the answer. How can I solve option A, using Vieta's Theorem? Viete theorem always works. But, if you don't have information about the solutions of the quadratic equation, the Viete theorem's use not always facilitates mental calculations. Option A is exactly the case.



Intern
Joined: 17 Jul 2016
Posts: 35

Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
31 Jul 2016, 10:11
if x^2=px+q
then x=.5p+_root(.25p^2+q) This is very convenient to use in place of the quadratic formula when the coefficient of the linear term is even.
a) x^2=2x+1 ; x=1+_root(1+1) b) x^2=2x1; x=1+_root(11) c) x^2=2x1; x=1+_root(11) d) x^2=2x+1; x=1+_root(1+1) The answer has to be d.



Director
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 742
Location: India
GPA: 3.64

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
26 Nov 2016, 05:57
Another method to solve this question is : x=\((b±\sqrt{(b^24ac)})/2a\) , using equation for finding roots of quadratic equation where b is coefficient of x, a is coef of x^2 and c is constant. Substituting values for option A gives x=\(1±\sqrt{2}\) Since we need root as \(1+\sqrt{2}\), b must be negative, with other coef same as A which is option D Answer D.
_________________
Please give kudos, if you like my post
When the going gets tough, the tough gets going...



Manager
Joined: 03 Jan 2017
Posts: 178

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
25 Mar 2017, 09:40
let's just calculate answer for each: 1) (2+8)/2 we see that we need negative b, so that the root could be (2+8)/2 D is fine



Manager
Joined: 20 Apr 2014
Posts: 100

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
26 Mar 2017, 09:18
Hi Bunuel
Could you please put your input here ? I can not get the concept or people are talking about something I already miss BR.



Director
Joined: 26 Aug 2016
Posts: 687
Location: India
Concentration: Strategy, Marketing
GMAT 1: 690 Q50 V33 GMAT 2: 700 Q50 V33 GMAT 3: 730 Q51 V38
GPA: 4
WE: Consulting (Consulting)

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
26 Mar 2017, 09:54
Use the formula roots = { b +/ sqRt(b^2  4ac) } / 2a



Math Expert
Joined: 02 Sep 2009
Posts: 47983

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
28 Mar 2017, 02:59



Manager
Joined: 23 Jan 2016
Posts: 211
Location: India
GPA: 3.2

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
09 Apr 2017, 07:34
fla wrote: Bunuel wrote: Which of the following equations has 1 + √2 as one of its roots?
A) x^2 + 2x – 1 = 0 B) x^2 – 2x + 1 = 0 C) x^2 + 2x + 1 = 0 D) x^2 – 2x – 1 = 0 E) x^2 – x – 1= 0 If x1 = 1 + √2 then x2 = 1  √2. By the Viete theorem: b = (x1 + x2) = (1 + √2 + 1  √2) = 2 c = x1 * x2 = (1 + √2)(1  √2) = 1  2 = 1 is the answer. how did you get 1  √2 as the second root?? should it not be (1+√2) or 1√2?? Bunuel, I would really appreciate your help here, im trying to solve this question using vieta's theorum. Thank you.



Intern
Joined: 02 Oct 2013
Posts: 6

Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
13 May 2017, 05:56
use this equation a^2b^2 = (ab)(a+b)
\(x^22x1 = x^22x+12\) = \((x1)^2\sqrt{2}^2 = (x1+\sqrt{2})(x1\sqrt{2})\)
thus, x = 1+\sqrt{2} and 1\sqrt{2}



Intern
Joined: 28 Dec 2010
Posts: 23

Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
26 Jul 2017, 13:16
This problem involves irrational roots of a quadratic equation:
For a quadratic equation: ax^2+bx+c = 0 where, the coefficients a,b and c are real.
If we have one root as m +√n than it will also have a conjugate root m √n where m, n are rational and n is not a perfect square.
According to the question, one root is 1+√2 than the other root will be its conjugate i.e. 1√2.
Now, if we know the (sum of the roots) and (product of the roots) we can create a quadratic equation as:
x^2  (sum of roots) x + product of the roots = 0
Sum of roots = (1+√2) + (1√2) = 2 Product of roots = (1+√2) * (1√2) = 1  (√2)^2 = 1
Quadratic Equation = x^2 2x+1 = 0 i.e. option D.



Manager
Joined: 08 Apr 2017
Posts: 83

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
07 Nov 2017, 16:49
Bunuel wrote: Which of the following equations has 1 + √2 as one of its roots?
A) x^2 + 2x – 1 = 0 B) x^2 – 2x + 1 = 0 C) x^2 + 2x + 1 = 0 D) x^2 – 2x – 1 = 0 E) x^2 – x – 1= 0 x = 1 + √2 x  1 = √2 Squaring both sides (x  1)^2 = (√2)^2 X^2 + 1  2x = 2 x^2 – 2x – 1 = 0 Option D



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12196
Location: United States (CA)

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
10 Jan 2018, 20:46
Hi All, While this prompt might look a bit 'scary', you can answer it without doing a lot of complex math (but you need to pay attention to what each of the 5 equations implies (and whether you can actually get a sum of 0 in the end or not). To start, we're told that (1 + √2) is a 'root' of one of those equations, which means that when you plug that value in for X and complete the calculation, you will get 0 as a result. We know that √2 is greater than 1, so (1 + √2) will be GREATER than 2 (it's actually a little greater than 2.4, but you don't have to know that to answer this question). So, when you plug that value into X^2 (which appears in all 5 answers), you get a value that is GREATER than 4. To get that "greater than 4" value down to 0, we have to subtract something.... Also keep in mind that... (squaring a value greater than 2) > (doubling that same value) So subtracting 2X from X^2 would NOT be enough to get us down to 0... we would ALSO need to subtract the 1.... With those ideas in mind, there's only one answer that matches.... Final Answer: GMAT assassins aren't born, they're made, Rich
_________________
760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com
Rich Cohen
CoFounder & GMAT Assassin
Special Offer: Save $75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/
***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************



Manager
Status: Turning my handicaps into assets
Joined: 09 Apr 2017
Posts: 125

Re: Which of the following equations has 1 + √2 as one of its roots?
[#permalink]
Show Tags
02 May 2018, 22:59
I just plugged the approximate value of 1+ √2 as 1+ approx 1.4= approx 2.4 and tested on answer choices. D) satisfies as 5. something  4.something  1 is approx 0.
_________________
If time was on my side, I'd still have none to waste......




Re: Which of the following equations has 1 + √2 as one of its roots? &nbs
[#permalink]
02 May 2018, 22:59



Go to page
1 2
Next
[ 21 posts ]



