Let $\alpha$ and $\beta$ be the roots of the equation $(x - a)(x - b) = c$,where $c \neq 0$. What are the roots of the equation $(x - \alpha)(x - \beta) + c = 0$?

What is the maximum value of the expression $\frac{x}{x^2 - 5x + 9}$ for all real values of $x$?

Let $f(x)$ be a quadratic expression such that $f(0)+f(1)=0$. If $f(-2)=0$,then

Let $p(x)$ be a quadratic polynomial such that $p(0) = 1$. If $p(x)$ leaves a remainder of $4$ when divided by $x - 1$ and a remainder of $6$ when divided by $x + 1$,then:

If $S = \{a \in R : |2a - 1| = 3[a] + 2\{a\}\}$,where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$,then $72 \sum_{a \in S} a$ is equal to:

For what value of $k$ will the equation $x^2 - (3k - 1)x + 2k^2 + 2k = 0$ have equal roots?