### Quadratic Equation Question 4

#### Question 4 - 25 January - Shift 1

Let

$S={\alpha: \log _2(9^{2 \alpha-4}+13)-\log _2(\frac{5}{2} \cdot 3^{2 \alpha-4}+1)=2}$.

Then the maximum value of $\beta$ for which the equation $x^{2}-2(\sum _{\alpha \in S} \alpha)^{2} x+\sum _{\alpha \in S}(\alpha+1)^{2} \beta=0$ has real roots, is

## Show Answer

#### Answer: 25

#### Solution:

#### Formula: Nature of Roots (ii)

$\log _2(9^{2 \alpha-4}+13)-\log _2(\frac{5}{2} \cdot 3^{2 \alpha-4}+1)=2$

$\Rightarrow \frac{9^{2 \alpha-4}+13}{\frac{5}{2} 3^{2 \alpha-4}+1}=4$

$\Rightarrow \alpha=2 \quad$ or $\quad 3$

$\sum _{\alpha \in S} \alpha=5$ and $\sum _{\alpha \in S}(\alpha+1)^{2}=25$

$\Rightarrow x^{2}-50 x+25 \beta=0$ has real roots

$\Rightarrow \beta \leq 25$

$\Rightarrow \beta _{\max }=25$