CAT 2024 Slot 3 QA Question 10: The sum of all distinct real values of $x$ that satisfy the equation $10^x + \dfrac{4}{10^x} = \dfrac{81}{2}$

Question

The sum of all distinct real values of $x$ that satisfy the equation $10^x + \dfrac{4}{10^x} = \dfrac{81}{2}$

Accepted Answer

Let $y = 10^x$. Since $10^x>0$, we have $y>0$.

$y + \dfrac{4}{y} = \dfrac{81}{2}$
$2y^2 - 81y + 8 = 0$

If the two positive roots are $y_1$ and $y_2$, then their product is:

$y_1y_2 = \dfrac{8}{2} = 4$

The corresponding $x$ values are $\log_{10} y_1$ and $\log_{10} y_2$.

$x_1 + x_2 = \log_{10}y_

The sum of all distinct real values of xxx that satisfy the equation 10x+410x=81210^x + \dfrac{4}{10^x} = \dfrac{81}{2}10x+10x4​=281​