Sum and product of roots of a quadratic equation

[asteroidfodder]

A problem states: Find the roots of the quadratic equation x[sup:1zokhfsm]2[/sup:1zokhfsm] - x + k = 0 and the value of the constant k given that one root of the equation is twice the other.

So, with the normal notation of ax[sup:1zokhfsm]2[/sup:1zokhfsm] + bx + c:

The roots are r and 2r. The sum equals -b/a, so 3r = 1 and r = 1/3, and 2r = 2/3.

Also the product is 2r[sup:1zokhfsm]2[/sup:1zokhfsm] or 2 * (1/3)[sup:1zokhfsm]2[/sup:1zokhfsm], so k = c/a = 2/9.

My answers for the roots are then 1/3 and 2/3. However the answer in the book for the roots is -1/3 and -2/3. The answer for k agrees with mine.

I would greatly appreciate some help understanding this. Thanks.

Hmm, I just did the following problem in the book which is almost identical and got answers that agree with the book. I am beginning to suspect that the book has an error.

 

[mmm4444bot]

Your book has the wrong sign on both of those roots.

If r is a root of a polynomial, then x - r is a factor of that polynomial.

Using your results: (x - 1/3)(x - 2/3) = x^2 - x + 2/9

Using the book's results: (x + 1/3)(x + 2/3) = x^2 + x + 2/9

Something to keep in mind: All math texts contain errors.