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How to choose the integer parameters a, b, c, d, m to the following equation has two integer solutions?

sqrt(a* x + b) +  sqrt(c*x + d) = m. 

For example, the equation sqrt(x +5 ) + sqrt(8-x)=5 has two solution x = -1 and x = 4 

sim_eq.mw

How many positive integer solutions (a,b,c) to 2a+3b+3c=2010

Dear friends, I encounter the following problem. Let k and i be two given positive integers such that k>=2i>=4. How to find all nonnegative integers tuples (x_1, x_2, ..., x_i) such that k-2i<=x_1+2x_2+...+ix_i<=k-i? The buildin command "isolve" seems not work. Thanks.

Why won't isolve show all the solutions here?

> a := 6*x+2*y >= 48:
> b := 6*x+2*y <= 60:
>
> with(plots):
> aa := implicitplot(a, x = 0 .. 10, y = 0 .. 5, filledregions = true, transparency = .5):
> bb := implicitplot(b, x = 0 .. 10, y = 0 .. 5, filledregions = true, transparency = .5):

display(aa,bb)

We can see the region of solutions.  But if we use isolve ...