Embibe Experts Solutions for Chapter: Quadratic Equations, Exercise 2: Level 2

There are a few integer values of $a$ such that $\frac{a^2-3a-3}{a-2}$ is an integer. Find the sum of all these integer values of $a$.

Let $n$ be a positive integer such that one of the roots of the quadratic equation $4x^2-\left(4\sqrt{3}+4\right)x+\sqrt{3}n-24=0$ is an integer. Find the value of $n$.

If $x=\frac{1}{2}\left(\sqrt[3]{2009}-\frac{1}{\sqrt[3]{2009}}\right)$ and $y={\left(x+\sqrt{1+x^2}\right)}^3,$ then find the sum of the digits of $y$

Let $S_n=n^2+20n+12, n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square?

Suppose that the two roots of the equation $\frac{1}{x^2-10x-29}+\frac{1}{x^2-10x-45}-\frac{2}{x^2-10x-69}=0$ are $\alpha$ and $\beta .$ Find the value of $(-\alpha \beta ).$

For how many real values of $a$ will $x^2+2ax+2008=0$ has two integer roots?

Natural numbers $k,l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2-kx+I=0$ then $a+\frac{1}{ b}$ and $b+\frac{1}{a}$ are the roots of $x^2-px+q=0.$ What is the sum of all possible values of $q$?

If ${\lambda }_1$ and ${\lambda }_2$ are the two values of $\lambda$, such that the roots $\alpha$ and $\beta$ of the quadratic equation, $\lambda \left(x^2-x\right)+x+5=0$ satisfy $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }+\frac{4}{5}=0,$ then $\frac{{\lambda }_1}{{\lambda }_2^2}+\frac{{\lambda }_2}{{\lambda }_1^2}$ is equal to