2002 AMC 10A Problems/Problem 14: Difference between revisions

Latest revision as of 15:09, 18 February 2009

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-== Problem ==
+#redirect [[2002 AMC 12A Problems/Problem 12]]
-The 2 roots of the quadratic <math>x^2 - 63x + k = 0</math> are both prime. How many values of k are there?
-<math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 4 \qquad \text{(E)}&</math>More than 4
-==Solution==
-Consider a general quadratic with the coefficient of <math>x^2</math> as one and the roots as r and s. It can be factored as <math>(x-r)(x-s)</math> which is just <math>x^2-(r+s)x+rs</math>. Thus, the sum of the roots is the negative of the coefficient of x and the product is the constant term. (In general, this leads to Vieta's Formulas).
-We now have that the sum of the two roots is 63 while the product is k. Since both roots are primes, one must be 2, otherwise the sum is even. That means the other root is 61 and the product must be 122. Hence, our answer is <math>\boxed{\text{(B)}\ 1 }</math>.