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

 It is currently 15 Oct 2019, 21:51 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

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.  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

Manager  B
Joined: 08 Sep 2016
Posts: 102
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

Show Tags

I found estimation to work the best.
Sqrt2= 1.4
1+sqrt2 = 2.4
One root = 2.4 or x =2.4
X^2 = 5.76
2x= 4.8

Now you have all the values to solve. You are trying to see which expression will give you the closest value to zero.

C can be immediately crossed off because you are adding throughout the expression

When you subtract 5.76 - 4.8, you get a value close to 1 being left over.

Equation D matches what you are looking for.
Manager  S
Joined: 10 Apr 2018
Posts: 130
GMAT 1: 600 Q44 V28 GPA: 3.56
WE: Engineering (Computer Software)
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

This is one of the simplest methods. Thanks.
_________________

The Graceful
----------------------------------------------------------
Every EXPERT was a beginner once...
Don't look at the clock. Do what it does, keep going
..
To achieve great things, two things are needed:a plan and not quite enough time - Leonard Bernstein.
Intern  Joined: 06 May 2018
Posts: 4
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

Show Tags

How can we use the sum of roots formula here?
We don't have the value of a to be certain?
Intern  B
Status: Studying for GMAT
Joined: 06 Jan 2015
Posts: 19
Location: United Kingdom
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

Show Tags

Looking through the answer responses, I have not heard of Viete theorem. I have also never heard of the rules mentioned by JeffTargetTestPrep (see below):

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.

But thank you for making light of these rules and theorem.

What I did and worked well for me, I used 1 + √2 and substitute into the answer choices. I started with answer choice C and made my way down and finally got the answer D- this can be done relatively quickly!

The idea is to ensure the left hand side (LHS) of the equation equals to the right hand side (RHS) of the equation, as follows:

From answer choice D, x^2 – 2x – 1 = 0

x^2 – 2x – 1 = 0
x^2 -2x = 1
(1 + √2)^2 - 2(1 + √2 ) = 1
(1+2+2√2) - (2 +2√2) =1
1+2+2√2 - 2 -2√2 =1

From here you can cancel +2√2 and -2√2

1+2-2 = 1

1 = 1

LHS = RHS

VP  V
Joined: 23 Feb 2015
Posts: 1255
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

Hi Bunuel
May I get your attention?
But who knows x2 = 1 - √2.?
_________________
“The heights by great men reached and kept were not attained in sudden flight but, they while their companions slept, they were toiling upwards in the night.”

Do you need official questions for Quant?
3700 Unique Official GMAT Quant Questions
------
SEARCH FOR ALL TAGS
GMAT Club Tests
VP  V
Joined: 23 Feb 2015
Posts: 1255
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

Show Tags

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 $$-2\sqrt{2}$$.

In the highlighted part, is the squared (^2) mistakenly put?
Quote:
(B) is (x - 1)^2 so its roots are 1 and 1.
(C) is (x + 1)^2 so its roots are -1 and -1.

Could you explain the quoted part?
_________________
“The heights by great men reached and kept were not attained in sudden flight but, they while their companions slept, they were toiling upwards in the night.”

Do you need official questions for Quant?
3700 Unique Official GMAT Quant Questions
------
SEARCH FOR ALL TAGS
GMAT Club Tests
VP  V
Joined: 23 Feb 2015
Posts: 1255
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

Show Tags

souvonik2k wrote:
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

souvonik2k
Could you explain a bit this one, please? thanks
_________________
“The heights by great men reached and kept were not attained in sudden flight but, they while their companions slept, they were toiling upwards in the night.”

Do you need official questions for Quant?
3700 Unique Official GMAT Quant Questions
------
SEARCH FOR ALL TAGS
GMAT Club Tests
VP  V
Joined: 23 Feb 2015
Posts: 1255
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

Show Tags

EMPOWERgmatRichC wrote:
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....

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

EMPOWERgmatRichC
Thanks for the extra-ordinary explanation.
But how do someone convinced that those equations are perfectly fine ( i mean- they are not randomly written)?
_________________
“The heights by great men reached and kept were not attained in sudden flight but, they while their companions slept, they were toiling upwards in the night.”

Do you need official questions for Quant?
3700 Unique Official GMAT Quant Questions
------
SEARCH FOR ALL TAGS
GMAT Club Tests
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15259
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

To start, NOTHING about an Official GMAT question is ever 'random'; every aspect of the question (including what appears in the 5 answer choices) was specifically chosen for a reason - and sometimes the reason is to present a pattern that you can take advantage of.

From your question, I assume that you're asking about how we know for sure that the each of those equations actually does have a solution (or multiple solutions). Since we're looking for just one thing - the equation that has (1 + √2) as a root - it doesn't really matter what the other 4 equations are (since they won't have that root, none of them will be the answer to the question that was asked).

GMAT assassins aren't born, they're made,
Rich
_________________
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

Show Tags

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 $$-2\sqrt{2}$$.

In the highlighted part, is the squared (^2) mistakenly put?
Quote:
(B) is (x - 1)^2 so its roots are 1 and 1.
(C) is (x + 1)^2 so its roots are -1 and -1.

Could you explain the quoted part?

Yes, it should be
When you put $$x = (1 + \sqrt{2})$$ in them, you will get $$(+2\sqrt{2})$$ term.
Basically, it means $$x^2 = ( 1 + \sqrt{2})^2 = 1 + 2 + 2*\sqrt{2}$$

Recall that
$$(a + b)^2 = a^2 + b^2 + 2ab$$
$$(a - b)^2 = a^2 + b^2 - 2ab$$

B) x^2 – 2x + 1 = 0
Look at the left side. It is (x - 1)^2 because when you expand it, you get x^2 - 2x + 1.
$$(x - 1)^2 = (x - 1)*(x - 1) = 0$$
So x = 1, 1

C) x^2 + 2x + 1 = 0
Similarly, (x +1)^2 = x^2 + 2x + 1
So here (x + 1)(x + 1) = 0
x = -1, -1
_________________
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 >
SVP  P
Joined: 03 Jun 2019
Posts: 1689
Location: India
Re: Which of the following equations has 1 + √2 as one of its roots?  [#permalink]

Show Tags

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

Asked: Which of the following equations has 1 + √2 as one of its roots?
r1 = 1 + √2
r2 = 1 - √2
Sum of roots = 2
Product of roots = (1 + √2) * (1 - √2) = 1-2 = -1

Equation of quadratic: -
x^2 - 2x - 1 = 0

IMO D
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."

Please provide kudos if you like my post. Kudos encourage active discussions.

My GMAT Resources: -

Efficient Learning
All you need to know about GMAT quant

Tele: +91-11-40396815
Mobile : +91-9910661622
E-mail : kinshook.chaturvedi@gmail.com Re: Which of the following equations has 1 + √2 as one of its roots?   [#permalink] 15 Sep 2019, 10:51

Go to page   Previous    1   2   [ 31 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  