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Can someone help me answering this exercise?

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Can someone help me answering this exercise?

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!!!
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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.
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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.
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by torres.oriana » Mon Jun 28, 2010 6:00 am
Thanks so much!!!
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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
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