If the equations $x^2 + m x + 9 = 0$, $x^2 + n x + 17 = 0$ and $x^2 + (m+n) x + 35 = 0$ have a common negative root, then the value of $(2 m + 3 n)$ is

The roots $\alpha$, $\beta$ of the equation $3x^2 + \lambda x - 1 = 0$, satisfy $\dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2} = 15$. The value of $(\alpha^3 + \beta^3)^2$, is

If $x$ and $y$ are real numbers such that $4x^2 + 4y^2 - 4xy - 6y + 3 = 0$, then the value of $(4x + 5y)$ is

Let $\alpha$ and $\beta$ be the two distinct roots of the equation $2x^2 - 6x + k = 0$, such that $(\alpha + \beta)$ and $\alpha \beta$ are the distinct roots of the equation $x^2 + px + p = 0$. Then, the value of $8(k - p)$ is

The equation $x^3 + (2r+1)x^2 + (4r−1)x+2=0$ has $-2$ as one of its roots. If the other two roots are real, what is the minimum possible non-negative integer value of $r$?

If $x$ and $y$ are real numbers such that $x^2 + (x - 2y - 1)^2 = -4y(x + y)$, then value of $x - 2y$?

The number of integer solutions of the equation

is:

If $p^2 + q^2 - 29 = 2pq - 20 = 52 - 2pq,$ then the difference between the maximum and minimum possible values of $p^3 - q^3$ is:

Let $k$ be the largest integer such that the equation $(x-1)^2+2 k x+11=0$ has no real roots. If $y$ is a positive real number, then the least possible value of $\frac{k}{4 y}+9 y$ is

A quadratic equation $x^2 + bx + c = 0$ has two real roots. If the difference between the reciprocals of the roots is $13\frac{1}{3}$, and the sum of the reciprocals of the squares of the roots is $59\frac{5}{9}$, then the largest possible value of $(b + c)$ is:

Let $a$ and $b$ be natural numbers. If $a^2+a b+a=14$ and $b^2+a b+b=28$ , then $(2 a+b)$ equals

Let $a, b, c$ be non-zero real numbers such that $b^2 \lt 4 a c$ , and $f(x)=a x^2+b x+c$ . If the set $S$ consists of al integers $m$ such that $f(m)\lt0$ , then the set $S$ must necessarily be

Let $f(x)$ be a quadratic polynomial in $x$ such that $f(x) \geq 0$ for all real numbers $x$. If $f(2)=0$ and $f(4)=6$, then $f(-2)$ is equal to

Let $r$ and $C$ be real numbers. If $r$ and $-r$ are roots of $5x^3 + cx^2 - 10x + 9 = 0$, then $c$ equals

The number of integer solutions of the equation $\left(x^2-10\right)^{\left(x^2-3 x-10\right)}=1$ is

If $(3+2 \sqrt{2})$ is a root of the equation $a x^2+b x+c=0$, and $(4+2 \sqrt{3})$ is a root of the equation $a y^2+m y+n=0$, where $a, b, c, m$ and $n$ are integers, then the value of $\left(\frac{b}{m}+\frac{c-2 b}{n}\right)$ is

Suppose $k$ is any integer such that the equation $2x^2 + kx + 5 = 0$ has no real roots and the equation $x^2 + (k-5)x + 1 = 0$ has two distinct real roots for $x$. Then, the number of possible values of $k$ is

If $r$ is a constant such that $|x^2 - 4x - 13| = r$ has exactly three distinct real roots, then the value of $r$ is

Consider the pair of equations: $x^2 - xy - x = 22$ and $y^2 - xy + y = 34$. If $x > y$, then $x - y$ equals

Suppose one of the roots of the equation $ax^2 - bx + c = 0$ is $2 + \sqrt{3}$, where $a$, $b$ and $c$ are rational numbers and $a \neq 0$. If $b = 3c$ then $|a|$ equals

The number of distinct real roots of the equation (x + $\frac{1}{x}$ ) 2 - 3(x + $\frac{1}{x}$ ) + 2 = 0 equals

How many distinct positive integer-valued solutions exist to the equation $(x^2 - 7x + 11)^{(x^2 - 13x + 42)} = 1$?

In how many ways can a pair of integers $(x, a)$ be chosen such that

Let f(x) = x² + ax + b and g(x) = f(x + 1) - f(x - 1). If f(x) ≥ 0 for all real x, and g(20) = 72, then the smallest possible value of b is

Let $m$ and $n$ be positive integers. If $x^2 + mx + 2n = 0$ and $x^2 + 2nx + m = 0$ have real roots, then the smallest possible value of $m + n$ is

The product of the distinct roots of $|x^2 - x - 6| = x + 2.$ is

The number of solutions of the equation $|x|(6x^2 + 1) = 5x^2$ is

The quadratic equation $x^2 + bx + c = 0$ has two roots $4a$ and $3a$, where $a$ is an integer. Which of the following is a possible value of $b^2 + c$?

What is the largest positive integer $n$ such that

is also a positive integer?

Let $A$ be a real number. Then the roots of the equation $x^2 - 4x - \log_{2} A = 0$ are real and distinct if and only if

If $u^2 + (u - 2v - 1)^2 = -4v(u + v)$, then what is the value of $u + 3v$?

If $a$ and $b$ are integers such that $2x^2 - ax + 2 > 0$ and $x^2 - bx + 8 \geq 0$ for all real numbers $x$, then the largest possible value of $2a - 6b$ is [TITA]

If $f_1(x) = x^2 + 11x + n$ and $f_2(x) = x$, then the largest positive integer $n$ for which the equation $f_1(x) = f_2(x)$ has two distinct real roots, is: (TITA)

If $x + 1 = x^2$ and $x > 0$, then $2x^4$ is:

The minimum possible value of the sum of the squares of the roots of the equation $x^2 + (a + 3)x - (a + 5) = 0$ is