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I want to find the numbers $a, b, c, d, m, n, p, q, r$ so that the equation has the form $$ \left |(a x + b) (c x + d)\right | + \left |m x + n\right | + p x^2 + q x + r=0 $$ has six solutions $-5,-3,-1,1,2,3$. I tried

Clear[f, a, b, c, d, m, n, p, q, r]
f[x_] := Abs[(a x + b)* (c x + d)] + Abs[m x + n] + p x^2 + q x + r;
FindInstance[{f[-1] == 0 && f[-5] == 0 && f[-3] == 0 && f[1] == 0 && 
   f[2] == 0 && f[3] == 0 && 1 <= a <= 10 && -100 <= b <= 100 && 
   1 <= c <= 10 && -100 <= d <= 100 && 
   1 <= m <= 10 && -100 <= n <= 100 && -10 <= p <= -1 && -100 <= q <= 
    100 && -50 <= r <= 100 && a > c && a c + p != 0 && 
   c a - p != 0 }, {a, b, c, d, m, n, p, q, r}, Reals]

I got

{{a -> 2, b -> -8, c -> 1, d -> -(23/13), m -> 32/13, n -> 64/13, p -> -(34/13), q -> 118/13, r -> -(240/13)}}

I got the answer about 20 minutes (or longer). How can I decrease the time?

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1 Answer 1

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The use of RealAbs instead of Abs in your f[x_] does the job.

Clear[f, a, b, c, d, m, n, p, q, r]
f[x_] := RealAbs[(a x + b)*(c x + d)] + RealAbs[m x + n] + p x^2 + 
 q x + r;FindInstance[{f[-1] == 0 && f[-5] == 0 && f[-3] == 0 && f[1] == 0 && 
f[2] == 0 && f[3] == 0 && 1 <= a <= 10 && -100 <= b <= 100 && 
1 <= c <= 10 && -100 <= d <= 100 && 
1 <= m <= 10 && -100 <= n <= 100 && -10 <= p <= -1 && -100 <= q <=
  100 && -50 <= r <= 100 && a > c && a c + p != 0 && 
c a - p != 0}, {a, b, c, d, m, n, p, q, r}, Reals]//AbsoluteTiming

{30.195,{{a->3/2,b->-(43/8),c->1,d->98/67,m->40685/4288,n->-(56959/4288),p->-(4921/2144),q->27045/4288,r->-(74397/4288)}}}

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