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Question
Math
Posted about 1 month ago

 若 m2=m+1n2=n+1 求 m+n=\begin{array}{c}\text { 若 } m^{2}=m+1 \quad n^{2}=n+1 \\ \text { 求 } m+n=\end{array}
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Answer from Sia
Posted about 1 month ago
Solution by Steps
step 1
Solve the equation m2=m+1m^2 = m + 1 for mm
step 2
The solutions to the quadratic equation m2m1=0m^2 - m - 1 = 0 are m=12(15)m = \frac{1}{2}(1 - \sqrt{5}) and m=12(1+5)m = \frac{1}{2}(1 + \sqrt{5})
step 3
Solve the equation n2=n+1n^2 = n + 1 for nn
step 4
The solutions to the quadratic equation n2n1=0n^2 - n - 1 = 0 are n=12(15)n = \frac{1}{2}(1 - \sqrt{5}) and n=12(1+5)n = \frac{1}{2}(1 + \sqrt{5})
step 5
Since mm and nn have the same solutions, we can write m+n=(12(15))+(12(1+5))m + n = \left(\frac{1}{2}(1 - \sqrt{5})\right) + \left(\frac{1}{2}(1 + \sqrt{5})\right)
step 6
Simplifying, we get m+n=12(15+1+5)=12(2)=1m + n = \frac{1}{2}(1 - \sqrt{5} + 1 + \sqrt{5}) = \frac{1}{2}(2) = 1
Answer
m+n=1m + n = 1
Key Concept
Solving quadratic equations
Explanation
The key concept here is solving quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0 using the quadratic formula x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

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