For how many values of is it true that:

(1) a is a positive integer such that a is less than or equal to 50.

(2) the quadratic equation x^2 + (2a+1)x + a^2 has two integer solutions?

Any help is appreciated. Thanks in advance!

Guest Jan 29, 2019

#1**+3 **

\(\text{I assume this is one question and that both must be satisfied}\\ x^2 + (2a+1)x + a^2 \text{ has solutions of }\\ x = \dfrac{-(2a+1)\pm \sqrt{(2a+1)^2 -4a^2}}{2}\\ \text{in order for these to be integers}\\ (2a+1)^2 -a^2 \text{ must the be the perfect square of an odd number}\)

\((2a+1)^2 - 4a^2 = \\ 4a^2 + 4a+1-4a^2 = 4a+1 \\ \text{and this must be the perfect square of an odd number if }x \text{ is to be an integer}\)

\(\text{The values of }a \text{ that satisfy this are }\\ (2,6,12,20,30,42) \\ \text{and all of these lead to 2 unique integer solutions of the given quadratic}\)

.Rom Jan 29, 2019