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3 problems match “Vieta's formulas”. Clear

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1. Sum of vertical asymptotes ●○○○○ 10 ptsThe graph of $y=\frac{5x^2-9}{3x^2+5x+2}$ has vertical asymptotes at $x = a$ and $x = b$. Find $a + b$. Round to two decimal places.
2. Ratio in a geometric sequence ●●●○○ 50 ptsIn the geometric sequence $\{a_n\}$ of real numbers, $a_5 \cdot a_{11} = 4$ and $a_3 + a_{13} = 5$. Find all possible values of $\frac{a_{14}}{a_4}$. Give each value as a decimal, in increasing order.
3. Longest chain of quadratic equations ●●●●○ 100 ptsQuadratic equations are solved one after another. Each equation has leading coefficient 1, of the form $x^2+p_jx+q_j=0$. Whenever an equation has two distinct real roots, the next equation is formed as follows: its constant term is the larger root of the current equation, and its linear coefficient is the smaller root of the current equation. The process stops at the first equation that does not have two distinct real roots. Find the maximum possible number of quadratic equations in such a sequence.