## Thursday, December 5, 2013

### Sum and Product of roots : Vieta -- > Questions from 2010-2011 Mathcounts Super Stretch

Questions: (detailed solutions below)

#1 : What is the sum of the solutions of 6x2 + 5x - 4 = 0 ? Express your answer as a common fraction.

#2 : A quadratic equation of the form x2 + kx + m = 0 has solutions x = 3 + 2 2  and 3 - 2 2
What is the value of k + m?

#3 : What is the sum of the reciprocals of the solutions of 4x2 - 13x + 3 = 0 ? Express your answer as a common fraction.

Solutions :

#1:  6x2 + 5x -4 = 0 divided the whole equation by 6 and you have x2 + (5/6) x - 4/6 = 0, which means that the sum of the solutions is - 5/6.

#2: The two roots are 3 + 2 2  and 3 - 2 2 , which means that -k = 3 + 2 2 + 3 - 2 2
k = -6;  m = (3 + 2 2 ) (3 - 2 2 ) = 9 - 8 = 1 so m + k = -6 + 1 = -5

#3:
Solution I:
4x2 - 13x + 3 = 0; divided the whole equation by 4 and you have  x2 - (13/4)x + 3/4 = 0,
which means that the sum of the two roots, if they are x and y, are 13/4 and their product is 3/4.

1/x + 1/y = (x + y) / xy = 13/4  divided by 3/4 = 13/3

Solution II: Tom shows Rob and Rob shows me how to solve this using another method.

The original equation is 4x2 -13x + 3 = 0 To find the sum and product of the reciprocals, you flip the equation so it becomes 3x2 - 13x + 4 = 0

Using the same way you find the sum of the two roots,the answer is 13/3.