Quadratic Equations

Let $P$ and $Q$ be the vertices of the parabolae $y=x^2+b x+c$ and $y=-x^2+d x+e$, respectively. If $P$ and $Q$ are the points of intersection of the parabolae then the slope of the line through $P$ and $Q$ is

Let $m$ and $p$ be real numbers such that the polynomial $f(x) = x^2 + mx + p$ has two distinct negative rational roots. Then the polynomial $g(x) = x^2 - (m^2 - 2p)x + p^2$ has distinct

A possible solution of the system of equations $x^2 - 8xy + 16y^2 = 0$ $(\log_{10} x)^2 + 2(\log_{10} x)(\log_{10} y) + (\log_{10} y)^2 = 4$ is

Let $f(x)=x^3-a x^2+b x+c$, where $a, b, c$ are nonzero integers. Suppose that $f(a)=0$. Then, all the roots of $f$ are integers if and only if

Let $S$ be the set of all rational numbers $r$ such that if $r=\frac{p}{q}$ with $p, q \in \mathbb{Z}, q \neq 0$, then the two solutions of the quadratic equation $x^2+2 p x+q^2=0$ are equal. Then the number of elements…

Let $f$ be a polynomial such that $f(x)=x^3+f^{\prime}(0) x^2+f^{\prime \prime}(1) x+6$ for all $x \in \mathbb{R}$. Then

Let $a, b$ be the roots of the polynomial $3 X^2+5 X-2$. Let $A$ be any $3 \times 3$ matrix whose entries are from the set $S=\{a, b\}$. Then

Let $f(X)=X^2+8 X+25$. Then the number of solutions in $\mathbb{R}$, of the equation $(f(X))^2-18 f(X)+80=0$ is

The number of positive integers $n$ such that the equation $n^2 X^2+2024 X+\left(n^2+\right.$ $1)=0$ has a solution in $\mathbb{R}$ is

For positive real numbers $a$ and $b$, the number of solutions of $a X^{887}-b=0$ in $\mathbb{R}$ is

The horizontal line $y=k$ intersects the parabola $y=2(x-4)(x-6)$ at points $A$ and $B$. If the length of $A B$ is 8 , then the value of $k$ is

The number of real roots of $f(x)=x^6+x^3-1$ is