The sum of all distinct real values of x
that satisfy the equation 10^x + 4/(10^x) = 17/2

Question

The sum of all distinct real values of x
 that satisfy the equation 10^x + 4/(10^x) = 17/2

UPSC Quizroom · Accepted Answer

Let y = 10^x (so y > 0)

Then:
y + 4/y = 17/2

Multiply by y:
y^2 + 4 = (17/2)·y

Multiply by 2:
2y^2 − 17y + 8 = 0

Solve quadratic:
Discriminant = (−17)^2 − 4·2·8 = 289 − 64 = 225
√225 = 15

y = (17 ± 15) / 4
So y = 8 or y = 1/2

Recall y = 10^x:

10^x = 8 → x = log10(8) = 3 log10(2)
10^x = 1/2 → x = log10(1/2) = − log10(2)

Sum of real x values = 3 log10(2) − log10(2) = 2 log10(2)

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