The sum of all the solutions to the equation $2 {log}_{10} x - {log}_{10} (2 x - 75) = 2$

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350

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200

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Solution

The correct option is D 200
$2 {log}_{10} x - {log}_{10} (2 x - 75) = 2$
$\Rightarrow {log}_{10} (x)^{2} - {log}_{10} (2 x - 75) = 2$
$\Rightarrow {log}_{10} [\frac{x^{2}}{(2 x - 75)}] = 2$
$\Rightarrow \frac{x^{2}}{(2 x - 75)} = 100$
$\Rightarrow x^{2} - 200 + 7500 = 0$ {\quadratic in x}
Sum of solution $= \frac{- b}{a} = \frac{- (- 200)}{1} = 200$

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