Let's assume that √2 + 3/√2 is a rational number. That means it can be expressed as a ratio of two integers a and b, where b is not equal to zero, and a and b have no common factors other than 1.
√2 + 3/√2 = a/b
Multiplying both sides by b√2 gives:
(√2 + 3/√2) * b√2 = a
Simplifying the left side of the equation:
b√2 * √2 + 3b = a
2b + 3b√2 = a
Rearranging the equation:
3b√2 = a - 2b
Squaring both sides:
18b^2 = a^2 - 4ab + 4b^2
a^2 - 4ab + 2b^2 = 18b^2
a^2 - 4ab - 16b^2 = 0
The left side of the equation is a quadratic expression in terms of a, and it can be solved using the quadratic formula:
a = [4b ± √(16b^2 + 64b^2)]/2
a = [4b ± 8b√2]/2
a = 2b ± 4b√2
We know that a and b have no common factors other than 1. However, if a is even, then 2 is a common factor of a and 2b, which contradicts our assumption. Therefore, a must be odd.
If we substitute a = 2n + 1, where n is an integer, into the equation a^2 - 4ab - 16b^2 = 0, we get:
(2n + 1)^2 - 4(2n + 1)b - 16b^2 = 0
4n^2 + 4n + 1 - 8nb - 4b - 16b^2 = 0
4n^2 + (4 - 8b)n + (1 - 4b - 16b^2) = 0
The discriminant of this quadratic equation is:
(4 - 8b)^2 - 4(4)(1 - 4b - 16b^2) = 32b^