F(x) = 8891x
2 + ax + 1988, and:
G(x) = 1988x
2 + ax + 8891
Find the value of a such that F(x) and G(x) contain a common zero.
Change terminology to avoid dealing with large integers:
F(x) = ax^2 + bx + c has roots r1,r2
G(x) = cx^2 + bx + a has roots r3,r4
s.t. the root named first is the more negative of the two roots.
With the change of variable names, we are trying to solve for b.
We can try setting r1=r3, r1=r4, r2=r3, or r2=r4.
Depending on the values for a and c, some setups may not have a real solution.
In particular r1=r3 or r2=r4 I believe require that a=c, so set the more positive root of one equal to the more negative root of the other.
Case1: c(-b - √(b^2 - 4ac)) = a(-b + √(b^2 - 4ac))
or
Case2: c(-b + √(b^2 - 4ac)) = a(-b - √(b^2 - 4ac))
Case1:
-bc + ab = (a+c)√(b^2 - 4ac)
b(a-c) = (a+c)√(b^2 - 4ac)
(a-c)^2 b^2 = (a+c)^2 b^2 - 4ac(a+c)^2
((a+c)^2 - (a-c)^2) b^2 = 4ac(a+c)^2
4ac b^2 = 4ac(a+c)^2
b^2 = (a+c)^2
b = ±(a+c) = ±10879
Case2: probably the same answer but did not work it out.
https://www.desmos.com/calculator/jyr0ixewqs
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Posted by Larry
on 2024-09-21 08:37:24 |
Probable solution
