Can someone help me answering this exercise?

Expert replies

torres.oriana: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Mon Jun 07, 2010 10:48 am; Location: Bogota

by torres.oriana » Sun Jun 27, 2010 6:25 pm

If 'a' and 'b' are the roots of the quadratic equation x2 - x + 1 = 0, what is the value of a2 + b2?
Note: x2 is x squared, and a2 and b2 ate a and b squared.

The answer is -1.

The source of the exercise is:

https://free-quiz.4gmat.com/quizzes/math ... on_3.shtml

=====

How the hell does one get there?
My logic tells me to find the roots using the ...

(x-a)(x-b)=0

... form, but I just can`t find the right numbers!! There are no integer numbers whose product is 1 and whose sum is -1 ... are there?

There must be other algrebraic-type of solution which I am not seeing!!

Thanks!!!

Like this post

Source: Beat The GMAT — Problem Solving | Sun Jun 27, 2010 6:25 pm

GMAT/MBA Expert

Rahul@gurome: GMAT Instructor; Posts: 1179; Joined: Sun Apr 11, 2010 9:07 pmFollowed by: 88 members; Location: Milpitas, CA; Thanked: 447 times

by Rahul@gurome » Sun Jun 27, 2010 6:50 pm

In a quadratic equation, if the roots are say p and q, then the negative of their sum which is -(p+q) is given by the coefficient of x and the product of the roots which is pq is given by the constant term.
For the quadratic equation given above, roots are a and b and coefficient of x is -1. Also constant term is 1.
So -(a+b) = -1 and ab = 1
Or a+b = 1 and ab = 1.
Now a^2+b^2 = (a+b)^2 - 2a*b = 1^2 - 2*1 = 1 - 2 = -1.

So algebraically, we are getting -1 as the answer.
Don't look for actual values because only imaginary numbers will satisfy the condition a^2+b^2= -1, because in real numbers squares are always positive and so their sum can never be negative.

Rahul Lakhani
Quant Expert
Gurome, Inc.
https://www.GuroMe.com
On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)

Like this post

Testluv: GMAT Instructor; Posts: 1302; Joined: Mon Oct 19, 2009 2:13 pmFollowed by: 164 members; Location: Toronto; Thanked: 539 times; GMAT Score:800

by Testluv » Sun Jun 27, 2010 7:59 pm

Hi,

it should be noted that imaginary numbers are outside the scope of the GMAT. On the GMAT, squares will always be positive or zero.

Kaplan Teacher in Toronto

Like this post

torres.oriana: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Mon Jun 07, 2010 10:48 am; Location: Bogota

by torres.oriana » Mon Jun 28, 2010 6:00 am

Thanks so much!!!

Like this post

kvcpk: Legendary Member; Posts: 1893; Joined: Sun May 30, 2010 11:48 pmFollowed by: 7 members; Thanked: 215 times

by kvcpk » Mon Jun 28, 2010 7:01 am

torres.oriana wrote:If 'a' and 'b' are the roots of the quadratic equation x2 - x + 1 = 0, what is the value of a2 + b2?
Note: x2 is x squared, and a2 and b2 ate a and b squared.

The answer is -1.

How the hell does one get there?

There must be other algrebraic-type of solution which I am not seeing!!

Thanks!!!

Hi torres,

Are you aware of soemthing called sum of the roots and product of the roots of a quadratic equation.
It will be useful to remember this.
For a quadratic equation of the form px^2+qx+r = 0, sum of the roots is -q/p and product of the roots is r/p

In this case, eqtn is x2 - x + 1 = 0
so p = 1, q = -1, r =1
therefore, a+b = -q/p = 1
ab = r/p = 1

we know that (a+b)^2 = a^2+b^2+2ab
so a^2+b^2 = 1-2(1) = -1

Hope this helps!!

Praveen

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

559 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review