• 5 Day FREE Trial
Study Smarter, Not Harder

Available with Beat the GMAT members only code

• Free Trial & Practice Exam
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Free Veritas GMAT Class
Experience Lesson 1 Live Free

Available with Beat the GMAT members only code

• 5-Day Free Trial
5-day free, full-access trial TTP Quant

Available with Beat the GMAT members only code

• Free Practice Test & Review
How would you score if you took the GMAT

Available with Beat the GMAT members only code

• Award-winning private GMAT tutoring
Register now and save up to \$200

Available with Beat the GMAT members only code

• 1 Hour Free
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Magoosh
Study with Magoosh GMAT prep

Available with Beat the GMAT members only code

• Get 300+ Practice Questions

Available with Beat the GMAT members only code

## OG Root Q

This topic has 10 expert replies and 0 member replies
AbeNeedsAnswers Master | Next Rank: 500 Posts
Joined
02 Jul 2017
Posted:
191 messages
Followed by:
1 members
Thanked:
1 times

#### OG Root Q

Wed Jul 19, 2017 10:08 pm
Elapsed Time: 00:00
• Lap #[LAPCOUNT] ([LAPTIME])
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

D

Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!

### GMAT/MBA Expert

Jay@ManhattanReview GMAT Instructor
Joined
22 Aug 2016
Posted:
703 messages
Followed by:
17 members
Thanked:
332 times
Wed Jul 19, 2017 11:59 pm
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

D
We see that one of the roots is (1 + √2), which is not a whole number or it is an irrational root, thus the other root of the quadratic equation must be the conjugate of (1 + √2) = (1 - √2).

To get the conjugate, reverse the sign of the irrational part. Since in the given root, (1 + √2), the sign of the irrational part is +, so we must make it - to get its conjugate.

Thus, the two roots are: (1 + √2) and (1 - √2)

You must know that any quadratic equation can be written as:

x^2 - (Sum of the roots)*x + (Product of the roots) = 0

For the given roots,

Sum of the roots = (1 + √2) + (1 - √2) = 2, and
Product of the roots = (1 + √2) x (1 - √2) = 1^2 - (√2)^2 = 1 - 2 = -1

Thus, the quadratic equation is x^2 - (2)*x + (-1) = 0

x^2 - 2x -1 = 0

Hope this helps!

-Jay
_________________
Manhattan Review GMAT Prep

Locations: New York | Bangkok | Abu Dhabi | Rome | and many more...

### GMAT/MBA Expert

Jay@ManhattanReview GMAT Instructor
Joined
22 Aug 2016
Posted:
703 messages
Followed by:
17 members
Thanked:
332 times
Thu Jul 20, 2017 12:10 am
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

D
Another approach could be plug-in the value. Though handling the square of (1 + √2) would be difficult, doing a step at a time and analyzing at each level would save time and take us home.

Let's take option A.

A. x^2 + 2x - 1 = 0

(1 + √2)^2 + 2(1 + √2) - 1 = 0

(1 + 2 + 2√2) + (2 + 2√2) -1 = 0

To get '0' on the RHS, every term must cancel; however, we see that only '1' gets canceled, while '2' and '2√2' gets added. To cancel them, the sign of the middle term (shown in blue) of the quadratic equation x^2 + 2x - 1 = 0 must be negative.

Thus, the equations should be x^2 - 2x - 1 = 0.

Let's verify...

x^2 - 2x - 1 = 0

(1 + √2)^2 - 2(1 + √2) - 1 = 0

1 + 2 + 2√2 - 2 - 2√2 - 1 = 0

0 = 0.

Thus, the correct answer is Option D.

Hope this helps!

-Jay
_________________
Manhattan Review GMAT Prep

Locations: New York | Bangkok | Abu Dhabi | Rome | and many more...

### GMAT/MBA Expert

Rich.C@EMPOWERgmat.com Elite Legendary Member
Joined
23 Jun 2013
Posted:
8699 messages
Followed by:
460 members
Thanked:
2729 times
GMAT Score:
800
Thu Jul 20, 2017 11:32 am

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....

GMAT assassins aren't born, they're made,
Rich

_________________
Contact Rich at Rich.C@empowergmat.com

### GMAT/MBA Expert

GMATGuruNY GMAT Instructor
Joined
25 May 2010
Posted:
13355 messages
Followed by:
1779 members
Thanked:
12877 times
GMAT Score:
790
Thu Jul 20, 2017 11:49 am
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
A, B, C and D imply the following:
A) x² + 2x = 1
B) x² - 2x = -1
C) x² + 2x = -1
D) x² - 2x = 1.

Since these options all include x² and 2x , calculate x² = (1 + √2)² and 2x = 2(1 + √2):
= (1 + √2)² = 1² + √2² + 2(1)(√2) = 3 + 2√2.
2x = 2(1 + √2) = 2 + 2√2.

If we subtract the blue expressions from the red expressions, we get:
x² - 2x = (3 + 2√2) - (2 + 2√2)
x² - 2x = 1.

_________________
Mitch Hunt
GMAT Private Tutor
GMATGuruNY@gmail.com
If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
Available for tutoring in NYC and long-distance.

Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.

### GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
Joined
12 Sep 2012
Posted:
2560 messages
Followed by:
113 members
Thanked:
581 times
Target GMAT Score:
V51
GMAT Score:
780
Sun Jul 23, 2017 3:33 pm
I've got a few ways to solve this, but I'll start by invoking my favorite, the Lazy Testwriter Principle.

If 1 + √2 is a root, then the laziest other root we could possibly dream up is 1 - √2. The sum of the roots is (1 + √2) + (1 - √2), or 2. The product of the roots is (1 + √2) * (1 - √2), or -1.

In a basic quadratic in which the first term is x², the quadratic form is x² + (-1)*(sum of roots)*x + (product of roots) = 0. Since the sum is 2 and the product is -1, the quadratic must be x² - 2x - 1, and D is our only choice.

Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now!

### GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
Joined
12 Sep 2012
Posted:
2560 messages
Followed by:
113 members
Thanked:
581 times
Target GMAT Score:
V51
GMAT Score:
780
Sun Jul 23, 2017 3:37 pm
Another approach: using the factor we have.

We know from the quadratic formula that the solutions to a quadratic of the form ax² + bx + c = 0 are

x = (-b ± √(b² - 4ac))/2a

I know that a = 1 (since there's no coefficient on x² in any of the answers) and that one value of x = 1 + √2, so I can plug those into my formula:

1 + √2 = (-b ± √(b² - 4c))/2

2 + 2√2 = -b ± √(b² - 4c)

Eyeballing that, it would make perfect sense if -b = 2 and √(b² - 4c) = 2√2. If -b = 2, then b = -2, so I can plug that into the second part:

√(b² - 4c) = 2√2

√((-2)² - 4c) = 2√2

√(4 - 4c) = 2√2

4 - 4c = 8

-4c = -4

c = -1

And from there I know the coefficients: a = 1, b = -2, and c = -1, so my quadratic is x² - 2x - 1 = 0.

Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now!

### GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
Joined
12 Sep 2012
Posted:
2560 messages
Followed by:
113 members
Thanked:
581 times
Target GMAT Score:
V51
GMAT Score:
780
Sun Jul 23, 2017 3:46 pm
We could also the factor that we have in a different way. If the roots of my quadratic are r and s, I know that

(x - r) * (x - s) = 0

I know that one of the roots, let's say r, is 1 + √2, so

(x - (1 + √2)) * (x - s) = 0

Looking at my answer choices, I notice that none of them have √2 anywhere, so my other root should contain -√2. With that in mind, let's say s = z - √2, and replace that:

(x - (1 + √2)) * (x - (z - √2)) = 0

Now foil:

x² - (1 + √2 + z - √2)x + (1 + √2)*(z - √2) = 0

x² - (1 + z)*x + (z + √2z - √2 - 2) = 0

Looking at the answers, I know that z must be 1, since I need the √2s to cancel in the last ( ). That gives me

x² - (1 + 1)*x + (1 - 2) = 0

or

x² - 2x - 1 = 0

Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now!

### GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
Joined
12 Sep 2012
Posted:
2560 messages
Followed by:
113 members
Thanked:
581 times
Target GMAT Score:
V51
GMAT Score:
780
Sun Jul 23, 2017 3:50 pm
A pretty crude option would be guesstimation. √2 is about 1.4, so our root is about 2.4.

Plugging 2.4 into the answers should give me one quadratic that actually works and four that don't.

A:: 2.4² + 2*2.4 - 1 = 0, no, left side way too positive
B:: 2.4² - 2*2.4 + 1 = 0, no, the first term is bigger than the second, so we're positive, and adding the last +1 to that keeps us positive
C:: 2.4² + 2*2.4 + 1 = 0, no, obviously the left side is much bigger
D:: 2.4² - 2*2.4 - 1 = 0, hmmm, seems close
E:: 2.4² - 2.4 - 1 = 0, no, the ² term is much bigger than the other two

So D looks best, and we're set.

Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now!

### GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
Joined
12 Sep 2012
Posted:
2560 messages
Followed by:
113 members
Thanked:
581 times
Target GMAT Score:
V51
GMAT Score:
780
Sun Jul 23, 2017 3:58 pm
One last approach, and probably the worst of the five, is to use the x² term to help out.

Working with the answers, we can rewrite all of them in terms of x²:

A) x² = 1 - 2x
B) x² = 2x - 1
C) x² = -2x - 1
D) x² = 1 + 2x
E) x² = x + 1

From here, compute x²: (1 + √2)² => 3 + 2√2. With that in mind, the question now becomes:

Quote:
If x = 1 + √2, which of the following is equal to 3 + 2√2?

A) 1 - 2x
B) 2x - 1
C) -2x - 1
D) 1 + 2x
E) x + 1
A and C are negative, so they're no good. B is 2 * (1 + √2) - 1, which is too small. But D works, so we're done!

Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now!

### GMAT/MBA Expert

Jeff@TargetTestPrep GMAT Instructor
Joined
09 Apr 2015
Posted:
323 messages
Followed by:
6 members
Thanked:
30 times
Tue Jul 25, 2017 11:02 am
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 the 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.

Alternate Solution:

We will determine which equation becomes zero when x = 1 + √2. Notice that each of the equations has x^2; so, let’s begin by calculating (1 + √2)^2:

(1 + √2)^2 = 1^2 + 2√2 + 2 = 3 + 2√2

Now, to cancel the term 2√2 in x^2 = 3 + 2√2, the coefficient of 1 + √2 must be -2; so, let’s calculate x^2 - 2x:

(1 + √2)^2 - 2(1 + √2) = 3 + 2√2 - 2 - 2√2 = 1

So, to get a zero when we substitute x = 1 + √2, the equation must be x^2 - 2x - 1.

_________________
Jeffrey Miller Head of GMAT Instruction

### Best Conversation Starters

1 Vincen 180 topics
2 lheiannie07 61 topics
3 Roland2rule 54 topics
4 ardz24 44 topics
5 VJesus12 14 topics
See More Top Beat The GMAT Members...

### Most Active Experts

1 Brent@GMATPrepNow

GMAT Prep Now Teacher

155 posts
2 Rich.C@EMPOWERgma...

EMPOWERgmat

105 posts
3 GMATGuruNY

The Princeton Review Teacher

101 posts
4 Jay@ManhattanReview

Manhattan Review

82 posts
5 Matt@VeritasPrep

Veritas Prep

80 posts
See More Top Beat The GMAT Experts