Quadratic Equations – 22 January 2025 (Shift 1)

Solution:

Given equation: $e^{5(\ln x)^2 + 3} = x^8$.

Taking natural logarithm ($\ln$) on both sides:

$$ 5(\ln x)^2 + 3 = \ln(x^8) $$

$$ 5(\ln x)^2 + 3 = 8 \ln x $$

Let $\ln x = t$. Then:

$$ 5t^2 – 8t + 3 = 0 $$

$$ 5t^2 – 5t – 3t + 3 = 0 \Rightarrow 5t(t-1) – 3(t-1) = 0 $$

$$ (5t-3)(t-1) = 0 \Rightarrow t = 1 \text{ or } t = \frac{3}{5} $$

So, $\ln x = 1 \Rightarrow x_1 = e^1$.

And $\ln x = \frac{3}{5} \Rightarrow x_2 = e^{3/5}$.

Product of solutions = $x_1 \cdot x_2 = e^1 \cdot e^{3/5} = e^{1 + 3/5} = e^{8/5}$.

Ans. (1)