I am solving a problem using
Reduce, and having a difficulty with understanding the answers I obtained. I was solving
Reduce[abcd[q] = 0, q] where
abcd[q_]=-0.003 + (800 a^3)/(3 q^3) - (400 a^2 Sqrt[(a^2 + k q)/q^2])/( 3 q^2) - (100 k (k/q^2 - (2 (a^2 + k q))/q^3))/( 3 Sqrt[(a^2 + k q)/q^2]) + (200 a^2 (k/q^2 - (2 (a^2 + k q))/q^3))/( 3 q Sqrt[(a^2 + k q)/q^2])
It gave me back with 4 solutions - two real, two imaginary numbers. I put $a=1/4$, $k=1/10$ in the solutions, and obtained following numbers:
-26.8555 - 43.5291i -26.8555 + 43.5291i 4.8813 48.2047
I put these numerical solutions back into
abcd[q] with $a=1/4$, $k=1/10$, and obtained the following values:
-0.00593796 + 5.4479*10^-6 i -0.00593796 - 5.4479*10^-6 i 0.0656495 -1.30104*10^-18
Here is my first problem:
- While the fourth solution looks sufficiently close to $0$, others do not really seem so. What's happening? Aren't they supposed to be solutions?
Perplexed, I tried to obtain using different approach: first, I put $a=1/4$, $k=1/10$ in
abcd[q] and then tried
Reduce[abcd[q] = 0, q]. This gave me back a unique solution that coincides with the fourth solution, $48.2047$.
To double check, I plotted
abcd[q] with parameters $a=1/4$, $k=1/10$ and domain $[0, 100]$. It displayed a monotone decreasing function that intersects zero at $q=48.2047$. At $q=4.8813$,
abcd[q] was in fact strictly positive, perhaps $0.0656495$ as calculated above. In sum, it strongly seems that $q=48.2047$ is the right answer.
Here comes my second question:
- It seems that $q=48.2047$ is the right answer. Why did I get the other three answers from