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The equation f(x) = 0 has eight distinct real solution f also satisfy f(4+x) = f(4-x). The sum of all the eight solution of f(x) = 0 is
- a)12
- b)16
- c)32
- d)15
Correct answer is option 'C'. Can you explain this answer?
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The equation f(x) = 0 has eight distinct real solution f also satisfy ...
The Problem Statement
We have the equation f(x) = 0 with eight distinct real solutions and the property f(4+x) = f(4-x). We need to find the sum of all eight solutions.
Symmetry in the Function
- The condition f(4+x) = f(4-x) indicates that the function is symmetric about x = 4.
- This means that if x is a solution, then (8 - x) must also be a solution since f(4+x) = f(4-x) implies that the solutions are paired around x = 4.
Understanding the Solutions
- Given that there are eight distinct real solutions, we can pair them as follows:
- If s1, s2, s3, and s4 are four distinct solutions, then their paired solutions would be (8-s1), (8-s2), (8-s3), and (8-s4).
- Thus, the solutions can be grouped into four pairs:
- (s1, 8-s1)
- (s2, 8-s2)
- (s3, 8-s3)
- (s4, 8-s4)
Calculating the Sum of Solutions
- The sum of each pair is:
- s1 + (8 - s1) = 8
- s2 + (8 - s2) = 8
- s3 + (8 - s3) = 8
- s4 + (8 - s4) = 8
- Therefore, the total sum of all eight solutions is:
- 8 + 8 + 8 + 8 = 32
Conclusion
- The sum of all eight distinct real solutions to the equation f(x) = 0 is 32, which corresponds to option C.
We have the equation f(x) = 0 with eight distinct real solutions and the property f(4+x) = f(4-x). We need to find the sum of all eight solutions.
Symmetry in the Function
- The condition f(4+x) = f(4-x) indicates that the function is symmetric about x = 4.
- This means that if x is a solution, then (8 - x) must also be a solution since f(4+x) = f(4-x) implies that the solutions are paired around x = 4.
Understanding the Solutions
- Given that there are eight distinct real solutions, we can pair them as follows:
- If s1, s2, s3, and s4 are four distinct solutions, then their paired solutions would be (8-s1), (8-s2), (8-s3), and (8-s4).
- Thus, the solutions can be grouped into four pairs:
- (s1, 8-s1)
- (s2, 8-s2)
- (s3, 8-s3)
- (s4, 8-s4)
Calculating the Sum of Solutions
- The sum of each pair is:
- s1 + (8 - s1) = 8
- s2 + (8 - s2) = 8
- s3 + (8 - s3) = 8
- s4 + (8 - s4) = 8
- Therefore, the total sum of all eight solutions is:
- 8 + 8 + 8 + 8 = 32
Conclusion
- The sum of all eight distinct real solutions to the equation f(x) = 0 is 32, which corresponds to option C.
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The equation f(x) = 0 has eight distinct real solution f also satisfy f(4+x) = f(4-x). The sum of all the eight solution of f(x) = 0 isa)12b)16c)32d)15Correct answer is option 'C'. Can you explain this answer?
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