[Math Revolution GMAT math practice question]
Which of the following equations is satisfied by x=1+√2?
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 - 2 = 0
Which of the following equations is satisfied by x=1+√2?
This topic has expert replies
- Max@Math Revolution
- Elite Legendary Member
- Posts: 3991
- Joined: Fri Jul 24, 2015 2:28 am
- Location: Las Vegas, USA
- Thanked: 19 times
- Followed by:37 members
Timer
00:00
Your Answer
A
B
C
D
E
Global Stats
Math Revolution
The World's Most "Complete" GMAT Math Course!
Score an excellent Q49-51 just like 70% of our students.
[Free] Full on-demand course (7 days) - 100 hours of video lessons, 490 lesson topics, and 2,000 questions.
[Course] Starting $79 for on-demand and $60 for tutoring per hour and $390 only for Live Online.
Email to : [email protected]
GMAT/MBA Expert
- Jay@ManhattanReview
- GMAT Instructor
- Posts: 3008
- Joined: Mon Aug 22, 2016 6:19 am
- Location: Grand Central / New York
- Thanked: 470 times
- Followed by:34 members
Since the options are quadratic equations and x = 1 + √2 is a linear equation, we must find a way to convert into a quadratic equation such that we have a term with x.Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
Which of the following equations is satisfied by x=1+√2?
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 - 2 = 0
So, we have x = 1 + √2
=> x - 1 = √2
Squaring both the sides, we get
(x - 1)^2 = (√2)^2
x^2 - 2x + 1 = 2
x^2 - 2x - 1 = 0
The correct answer: A
Hope this helps!
-Jay
_________________
Manhattan Review GMAT Prep
Locations: Manhattan Review Chennai | Hyderabad | GRE Prep New Delhi | Tarnaka GRE Coaching | and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
- GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
Alternate approach:Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
Which of the following equations is satisfied by x=1+√2?
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 - 2 = 0
For any quadratic in the form x² + bx + c = 0, where b and c are rational numbers:
If one of the roots is m+√n, then the other root is m-√n.
Sum of the roots = -b
Product of the roots = c
Here, since one of the roots is 1+√2, the other root is 1-√2.
Thus:
-b = sum of the roots
-b = (1+√2) + (1-√2)
-b = 2
b = -2
c = product of the roots
c = (1+√2)(1-√2)
c = 1-2
c = -1
Resulting quadratic:
x² - 2x -1 = 0
The correct answer is A.
Last edited by GMATGuruNY on Tue Dec 11, 2018 10:06 am, edited 1 time in total.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
-
- Legendary Member
- Posts: 2214
- Joined: Fri Mar 02, 2018 2:22 pm
- Followed by:5 members
Looking at option B and D
$$B=x^2-2x+1:\ its\ roots\ 1\ and\ 1$$
because $$\left(x-1\right)^2=x^2-2x+1=0\ $$
$$D=x^2+2x+1=0;\ its\ root\ are\ -1\ and\ -1$$
because $$\left(x+1\right)^2=x^2+2x+1=0$$
If we substitute
$$1+\sqrt{2}\ into\ these\ equation,\ it\ is\ne0$$
The remaining 3 option have
$$x^2\ where\ \ x=1+\sqrt{2}=\left(1+\sqrt{2}\right)^2=2\sqrt{2}$$
This means that for
$$1+\sqrt{2}$$ to satisfy any of these equation must be =0 thus, $$x^2$$ which is $$2\sqrt{2}$$ to get 0
Option A is the only equation that satisfy these criteria.
$$x=1+\sqrt{2}$$
$$x-1=\sqrt{2}$$
$$x^2-2x-1=0$$
$$2\sqrt{2}-2\left(x-1\right)=0$$
$$2\sqrt{2}-2\sqrt{2}=0$$
$$OR\ \ \ x=1+\sqrt{2}\ \ \ $$
$$x-1=\sqrt{2}$$
$$square\ both\ sides\ \ \left(x-1\right)^2=\left(\sqrt{2}\right)^2$$
$$\left(x-1\right)\left(x-1\right)=2$$
$$x^2-x-x+1=2$$
$$x^2-2x+1=2$$
$$x^2-2x+1-2=0$$
$$x^2-2x-1=0\ \ $$
$$answer\ is\ Option\ A$$
$$B=x^2-2x+1:\ its\ roots\ 1\ and\ 1$$
because $$\left(x-1\right)^2=x^2-2x+1=0\ $$
$$D=x^2+2x+1=0;\ its\ root\ are\ -1\ and\ -1$$
because $$\left(x+1\right)^2=x^2+2x+1=0$$
If we substitute
$$1+\sqrt{2}\ into\ these\ equation,\ it\ is\ne0$$
The remaining 3 option have
$$x^2\ where\ \ x=1+\sqrt{2}=\left(1+\sqrt{2}\right)^2=2\sqrt{2}$$
This means that for
$$1+\sqrt{2}$$ to satisfy any of these equation must be =0 thus, $$x^2$$ which is $$2\sqrt{2}$$ to get 0
Option A is the only equation that satisfy these criteria.
$$x=1+\sqrt{2}$$
$$x-1=\sqrt{2}$$
$$x^2-2x-1=0$$
$$2\sqrt{2}-2\left(x-1\right)=0$$
$$2\sqrt{2}-2\sqrt{2}=0$$
$$OR\ \ \ x=1+\sqrt{2}\ \ \ $$
$$x-1=\sqrt{2}$$
$$square\ both\ sides\ \ \left(x-1\right)^2=\left(\sqrt{2}\right)^2$$
$$\left(x-1\right)\left(x-1\right)=2$$
$$x^2-x-x+1=2$$
$$x^2-2x+1=2$$
$$x^2-2x+1-2=0$$
$$x^2-2x-1=0\ \ $$
$$answer\ is\ Option\ A$$
- Max@Math Revolution
- Elite Legendary Member
- Posts: 3991
- Joined: Fri Jul 24, 2015 2:28 am
- Location: Las Vegas, USA
- Thanked: 19 times
- Followed by:37 members
=>
x=1+√2
=> x - 1 = √2
=> (x-1)^2 = (√2)^2
=> x^2 - 2x + 1 = 2
=> x^2 - 2x - 1 = 0.
Therefore, the answer is A.
Answer: A
x=1+√2
=> x - 1 = √2
=> (x-1)^2 = (√2)^2
=> x^2 - 2x + 1 = 2
=> x^2 - 2x - 1 = 0.
Therefore, the answer is A.
Answer: A
Math Revolution
The World's Most "Complete" GMAT Math Course!
Score an excellent Q49-51 just like 70% of our students.
[Free] Full on-demand course (7 days) - 100 hours of video lessons, 490 lesson topics, and 2,000 questions.
[Course] Starting $79 for on-demand and $60 for tutoring per hour and $390 only for Live Online.
Email to : [email protected]