• NEW! FREE Beat The GMAT Quizzes
Hundreds of Questions Highly Detailed Reporting Expert Explanations
• 7 CATs FREE!
If you earn 100 Forum Points

Engage in the Beat The GMAT forums to earn
100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Which of the following are roots of an equation 10x^{-2}+x^{ tagged by: Max@Math Revolution ##### This topic has 3 expert replies and 1 member reply ### GMAT/MBA Expert ## Which of the following are roots of an equation 10x^{-2}+x^{ ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult [Math Revolution GMAT math practice question] Which of the following are roots of an equation 10x^{-2}+x^{-1}-21=0 A. -2/3 or 5/7 B. -2/3 or 7/5 C. 2/3 or -5/7 D. 2/3 or -7/5 E. 3/2 or 5/7 _________________ Math Revolution Finish GMAT Quant Section with 10 minutes to spare. The one-and-only Worldâ€™s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. Only$149 for 3 month Online Course
Free Resources-30 day online access & Diagnostic Test
Email to : info@mathrevolution.com

### GMAT/MBA Expert

GMAT Instructor
Joined
09 Oct 2010
Posted:
1449 messages
Followed by:
32 members
59
Max@Math Revolution wrote:
[Math Revolution GMAT math practice question]

Which of the following are the roots of the equation 10x^{-2}+x^{-1}-21=0 ?

A. -2/3 or 5/7
B. -2/3 or 7/5
C. 2/3 or -5/7
D. 2/3 or -7/5
E. 3/2 or 5/7
$$?\,\,\,:\,\,\,\,{{10} \over {{x^2}}} + {1 \over x} - 21 = 0\,\,\,{\rm{roots}}$$

We can just substitute x with the values presented in the alternative choices. "Luckily" (A) will already be the correct answer.

Another approach, probably even faster (at least when the correct alternative choice is not the first!), is the following:

$${{10} \over {{x^2}}} + {1 \over x} - 21 = 0\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{{\rm{when}}\,\,x\,\, \ne \,0} \,\,\,10 + x - 21{x^2} = 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,21{x^2} - x - 10 = 0$$
$$21{x^2} - x - 10 = 0\,\,\,{\rm{has}}\,\,{\rm{product}}\,\,{\rm{of}}\,\,{\rm{roots}}\,\,\,{\rm{ = }}\,\,\, - {{10} \over {21}}\,\,\,\,\,\left( {{c \over a}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{ \,\left( A \right)\,\,{\rm{survives}} \hfill \cr \,\left( B \right)\,\,{\rm{out}} \hfill \cr \,\left( C \right)\,\,{\rm{survives}} \hfill \cr \,\left( D \right)\,\,{\rm{out}} \hfill \cr \,\left( E \right)\,\,{\rm{out}} \hfill \cr} \right.$$
$$21{x^2} - x - 10 = 0\,\,\,{\rm{has}}\,\,{\rm{sum}}\,\,{\rm{of}}\,\,{\rm{roots}}\,\,\,{\rm{ = }}\,\,\, - {{ - 1} \over {21}} = {1 \over {21}}\,\,\,\,\left( { - {b \over a}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{ \,\left( A \right)\,\, - {2 \over 3}{\rm{ + }}{5 \over 7}{\rm{ = }} - {{14} \over {21}} + {{15} \over {21}} = {1 \over {21}}\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,\left( A \right) \hfill \cr \,\,\left( C \right)\,\,{\rm{donÂ´t }}\,\,{\rm{need}}\,\,{\rm{checking}}\,\,\,\,\,\left( { - {5 \over 7} + {2 \over 3} = - {{15} \over {21}} + {{14} \over {21}} < 0 \,\,{\rm{out}}} \right) \hfill \cr} \right.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

### Top Member

Legendary Member
Joined
02 Mar 2018
Posted:
977 messages
Followed by:
2 members
$$10x^{\left(-2\right)}+x^{\left(-1\right)}-21=0$$
$$\frac{10}{x^2}+\frac{1}{x}-21=0$$
$$10+x-21x^2=0$$
$$-21x^2+x+10=0$$
$$-21x^2+15x-14x+10=0$$
$$-3x\left(7x-5\right)-2\left(7x-5\right)=0$$
$$\left(7x-5\right)\left(-3x-2\right)=0$$
$$7x-5=0$$ or $$-3x-2=0$$
$$\frac{7x}{7}=\frac{5}{7}$$ or $$\frac{-3x}{3}=\frac{2}{3}$$
$$x=\frac{5}{7}\ or\ x=-\frac{2}{3}$$
$$answer\ is\ option\ A$$

### GMAT/MBA Expert

GMAT Instructor
Joined
25 Apr 2015
Posted:
2456 messages
Followed by:
18 members
43
Max@Math Revolution wrote:
[Math Revolution GMAT math practice question]

Which of the following are roots of an equation 10x^{-2}+x^{-1}-21=0

A. -2/3 or 5/7
B. -2/3 or 7/5
C. 2/3 or -5/7
D. 2/3 or -7/5
E. 3/2 or 5/7
Multiplying the equation by x^2, we have:

10 + x - 21x^2 = 0

Multiplying the equation by -1, we have:

21x^2 - x - 10 = 0

(3x + 2)(7x - 5) = 0

3x + 2 = 0 â†' 3x = -2 â†' x = -2/3

or

7x - 5 = 0 â†' 7x = 5 â†' x = 5/7

_________________

Scott Woodbury-Stewart
Founder and CEO
scott@targettestprep.com

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

### GMAT/MBA Expert

Legendary Member
Joined
24 Jul 2015
Posted:
2240 messages
Followed by:
32 members
19
GMAT Score:
=>

10x^{-2}+x^{-1}-21=0
=> 10+x-21x^2=0 by multiplying x^2
=> 21x^2-x-10=0 by multiplying (-1)
=> (3x+2)(7x-5)=0
=> x = -2/3 or x = 5/7

_________________

Math Revolution
Finish GMAT Quant Section with 10 minutes to spare.
The one-and-only Worldâ€™s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
Only $149 for 3 month Online Course Free Resources-30 day online access & Diagnostic Test Unlimited Access to over 120 free video lessons-try it yourself Email to : info@mathrevolution.com • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE 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 • Magoosh Study with Magoosh GMAT prep 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 • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE 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

• 5 Day FREE Trial
Study Smarter, Not Harder

Available with Beat the GMAT members only code

• 1 Hour Free
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• FREE GMAT Exam
Know how you'd score today for \$0

Available with Beat the GMAT members only code

### Top First Responders*

1 Jay@ManhattanReview 77 first replies
2 Brent@GMATPrepNow 66 first replies
3 GMATGuruNY 33 first replies
4 Ian Stewart 24 first replies
5 Scott@TargetTestPrep 16 first replies
* Only counts replies to topics started in last 30 days
See More Top Beat The GMAT Members

### Most Active Experts

1 Scott@TargetTestPrep

Target Test Prep

200 posts
2 Max@Math Revolution

Math Revolution

94 posts
3 Brent@GMATPrepNow

GMAT Prep Now Teacher

92 posts
4 Jay@ManhattanReview

Manhattan Review

83 posts
5 GMATGuruNY

The Princeton Review Teacher

57 posts
See More Top Beat The GMAT Experts