Determine two distinct real numbers a and b that satisfy this system of equations:
a
2-b = 73
b
2-a =73
Note the number 73 is between 8^2 and 9^2. Specifically 64+9=73 and 81-8=73. So by inspection two solutions to (a,b) are (8,-9) and (-9,8).
A solution of two conics can have up to 4 real solutions.
Solving the second equation for a and substituting this into the first yields the quartic b^4-146b^2-b+5256.
Using synthetic division to divide out the known solutions reduces this to the quadratic
b^2-b-73=0
which is easily solved
b=(1 +/- sqrt(293))/2
Final two solutions
((1+sqrt(293))/2,(1+sqrt(293))/2)
((1-sqrt(293))/2,(1-sqrt(293))/2)
Edit: these last two don't count, since the problem states a,b are distinct.
Edited on July 25, 2024, 1:26 pm
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Posted by Jer
on 2024-07-23 09:19:48 |
Solution
