Exploring Math Olympiad Unique Answer Two Substitutions And A Clever Discriminant Argument
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- (9+1/x)⁴-(7+1/x)⁴=1040 → Find x Midpoint of 9+1/x and 7+1/x is 8+1/x. Let y=8+1/x. Then: 9+1/x=y+1 and 7+1/x=y-1. Equation: ...
- x³+x² = 1/8 → Find all real x The 1/8 is the tell. Let y=2x. Substituting x=y/
- x²-x-12=√(12x+48) → Find all real x. No
- 7^x+7^(2x)+7^(3x)=14 → Find x Let y=7^x. Then 7^(2x)=y² and 7^(3x)=y³. Equation: y+y²+y³=14. Rearrange: y³+y²+y-14=0.
- (x+4)-√(x-4)=2x-8 → Find all real x Let a=√(x+4) and b=√(x-4).
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2 Radical a+b+c=3 and 1/(a+1)+1/(b+1)+1/(c+1)=1 → Find abc Let x=a+1, y=b+1, z=c+1. First equation: x+y+z=6. (x+
(x²-
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