The sum of all distinct real values of x that satisfy the equation 10^x+4/10^x=81/2, is
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Explanatory Answer
Solution:
We are given the equation:
10^x + 4/(10^x) = 17/2
Step 1: Let 10^x = y
So, the equation becomes:
y + 4/y = 17/2
Multiply through by y:
2y^2 - 17y + 8 = 0
Step 2: Solve quadratic equation.
2y^2 - 17y + 8 = 0
Factorizing:
(2y - 1)(y - 8) = 0
So, y = 8 or y = 1/2
Step 3: Back-substitute.
10^x = 8 → x = log10(8) = log10(2^3) = 3 log10(2)
10^x = 1/2 → x = log10(1/2) = -log10(2)
Step 4: Sum of all possible values.
= 3 log10(2) + (-log10(2))
= 2 log10(2)
Final Answer:
2 log10(2)
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