BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Quadratic equation

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Newbie | Next Rank: 10 Posts
Posts: 7
Joined: Mon Jul 29, 2013 11:44 am
Thanked: 1 times

Quadratic equation

by kanika123 » Tue Jul 30, 2013 7:22 am
What can you say about the roots of px^2+qx+p=0?

A. One root is double the other.

B. One root is the square of the other.

C. One root is the reciprocal of the other.

D. The roots are equal in magnitudes but differ in signs.

E. None of These

How to solve this question in the fastest way possible?
Top
Source: — Problem Solving |
GMAT Instructor
Posts: 2630
Joined: Wed Sep 12, 2012 3:32 pm
Location: East Bay all the way
Thanked: 625 times
Followed by:119 members
GMAT Score:780

by Matt@VeritasPrep » Tue Jul 30, 2013 7:59 am
This doesn't really feel like a GMAT question to me -- it isn't unfair, per se, but the use of 'magnitude' is suspect: this might mean absolute value, in which the case the GMAT would say so, or it might (and generally does) mean the 'relative size' of a complex number (the 'distance' between that number and the origin of the complex plane), in which case the GMAT wouldn't use it at all.

That said, to solve this we can use the formulas for the sum of the roots and the product of the roots (also somewhat unusual for the GMAT) of a quadratic. If our quadratic is of the form ax^2 + bx + c = 0, the sum of the roots is (-b)/a and the product is (c/a). So our quadratic gives

Sum of roots = (-q/p)
Product of roots = (p/p) = 1

So if one of the roots is r, the other root must be 1/r, as r * (1/r) = 1. So the roots are reciprocal.

You could also solve this using the quadratic formula. The roots are (-q +/- √(q^2 - 4p^2))/(2p). The product of these roots is 1, so the roots are reciprocal.
Top
User avatar
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

by GMATGuruNY » Tue Jul 30, 2013 8:39 am
kanika123 wrote:What can you say about the roots of px²+qx+p=0?
An alternate approach is to plug in quadratics that satisfy the answer choices.
When the correct answer choice is satisfied, the coefficient of x² will be equal to the last term.

A. One root is double the other.
(x+2)(x+1) = 0 --> roots are x=-2 and x=-1.
(x+2)(x+1) = x² + 3x + 2.
Since the coefficient of x² is not equal to the last term, eliminate A.

B. One root is the square of the other.
(x-2)(x-4) = 0 --> roots are x=2 and x=4.
(x-2)(x-4) = x² - 6x + 8.
Since the coefficient of x² is not equal to the last term, eliminate B.

C. One root is the reciprocal of the other.
(x+2)(x + 1/2) = 0 --> roots are x=-2 and x = -1/2.
(x+2)(x + 1/2) = x² + (1/2)x + 2x + 1.
Success!
The coefficient of x² is equal to the last term.

To confirm, test one more random case:
(3x-1)(x-3) --> roots are x = 1/3 and x = 3.
(3x-1)(x-3) = 3x² - 9x - x + 3.
Success again!
The coefficient of x² is equal to the last term.

The cases above illustrate that answer choice C must be true.

The correct answer is C.
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
Top
Post new topic Post Reply
• Page 1 of 1