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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
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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.

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$$10x^{\left(-2\right)}+x^{\left(-1\right)}-21=0$$
$$\frac{10}{x^2}+\frac{1}{x}-21=0$$
$$10+x-21x^2=0$$
$$-21x^2+x+10=0$$
$$-21x^2+15x-14x+10=0$$
$$-3x\left(7x-5\right)-2\left(7x-5\right)=0$$
$$\left(7x-5\right)\left(-3x-2\right)=0$$
$$7x-5=0$$ or $$-3x-2=0$$
$$\frac{7x}{7}=\frac{5}{7}$$ or $$\frac{-3x}{3}=\frac{2}{3}$$
$$x=\frac{5}{7}\ or\ x=-\frac{2}{3}$$
$$answer\ is\ option\ A$$

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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

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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

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