Question

the sum of all the solutions of the equation

Answer 1

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Answer 2

Answer:

200

Step-by-step explanation:

The given equation is

$2log_{10}x-log_{10}(2x-75)=2$

Using the properties of logarithm, we get

$log_{10}x^2-log_{10}(2x-75)=2$        $[\because log a^b=blog a]$

$log_{10}\dfrac{x^2}{2x-75}=2$        $[\because log(\frac{a}{b})=log a-log b]$

$\dfrac{x^2}{2x-75}=10^2$        $[\because log_ax=y\Rightarrow x=a^y]$

$\dfrac{x^2}{2x-75}=100$

On cross multiplication we get

$x^2=100(2x-75)$

$x^2=200x-7500$

$x^2-200x+7500=0$

If a quadratic equation is $ax^2+bx+c=0$, then the sum of the solutions is -b/c.

Here, a=1 and b=-200. So, the sum of solutions is

$-\dfrac{b}{a}=-\dfrac{-200}{1}=200$

Therefore, the sum of all the solutions of the equation is 200.