GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 20 May 2019, 17:48 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. ### Request Expert Reply # Which of the following equations has 1 + √2 as one of its roots?

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 55188
Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

10
81 00:00

Difficulty:   55% (hard)

Question Stats: 64% (01:43) correct 36% (01:55) wrong based on 1462 sessions

### HideShow timer Statistics

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

_________________
##### Most Helpful Expert Reply
Target Test Prep Representative G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2823
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

25
32
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
_________________

# Jeffrey Miller

Head of GMAT Instruction

Jeff@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

##### Most Helpful Community Reply
Intern  Joined: 03 Jun 2015
Posts: 3
Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

3
13
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.
##### General Discussion
Manager  B
Joined: 07 May 2015
Posts: 175
Location: India
GMAT 1: 660 Q48 V31 GPA: 3
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

2
1
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 D
Joined: 16 Oct 2010
Posts: 9224
Location: Pune, India
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

9
8
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

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
Manager  Joined: 06 Jun 2014
Posts: 86
Location: United States
Concentration: Finance, General Management
GMAT 1: 450 Q27 V21 GPA: 3.47
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

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

1
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

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.
Manager  B
Joined: 17 Jul 2016
Posts: 56
Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

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=2x-1; x=1+_root(1-1)
c) x^2=-2x-1; x=-1+_root(1-1)
d) x^2=2x+1; x=1+_root(1+1)
The answer has to be d.
Ask GMAT Experts Forum Moderator V
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 1012
Location: India
GPA: 3.64
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

4
3
Another method to solve this question is :
x=$$(-b±\sqrt{(b^2-4ac)})/2a$$ , 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  S
Joined: 03 Jan 2017
Posts: 144
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

1
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  B
Joined: 20 Apr 2014
Posts: 88
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

1
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  P
Joined: 26 Aug 2016
Posts: 618
Location: India
Concentration: Operations, International Business
GMAT 1: 690 Q50 V33 GMAT 2: 700 Q50 V33 GMAT 3: 730 Q51 V38 GPA: 4
WE: Information Technology (Consulting)
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

1
Use the formula roots = { -b +/- sqRt(b^2 - 4ac) } / 2a
Math Expert V
Joined: 02 Sep 2009
Posts: 55188
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

1
5
hatemnag wrote:
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.

Check the links below:

Factoring Quadratics: http://www.purplemath.com/modules/factquad.htm
Solving Quadratic Equations: http://www.purplemath.com/modules/solvquad.htm

Theory on Algebra: http://gmatclub.com/forum/algebra-101576.html
Algebra - Tips and hints: http://gmatclub.com/forum/algebra-tips- ... 75003.html

DS Algebra Questions to practice: http://gmatclub.com/forum/search.php?se ... &tag_id=29
PS Algebra Questions to practice: http://gmatclub.com/forum/search.php?se ... &tag_id=50

Hope it helps.
_________________
Manager  S
Joined: 23 Jan 2016
Posts: 181
Location: India
GPA: 3.2
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

1
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  B
Joined: 02 Oct 2013
Posts: 5
Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

use this equation a^2-b^2 = (a-b)(a+b)

$$x^2-2x-1 = x^2-2x+1-2$$ = $$(x-1)^2-\sqrt{2}^2 = (x-1+\sqrt{2})(x-1-\sqrt{2})$$

thus, x = 1+\sqrt{2} and 1-\sqrt{2}
Intern  B
Joined: 28 Dec 2010
Posts: 23
Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

3
2
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  G
Joined: 08 Apr 2017
Posts: 77
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

7
3
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 V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 14174
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

8
2
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

*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****

# Rich Cohen

Co-Founder & GMAT Assassin Follow
Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/
Manager  B
Status: Turning my handicaps into assets
Joined: 09 Apr 2017
Posts: 128
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

### Show Tags

1
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?   [#permalink] 02 May 2018, 22:59

Go to page    1   2    Next  [ 23 posts ]

Display posts from previous: Sort by

# Which of the following equations has 1 + √2 as one of its roots?

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.

#### MBA Resources  