Found problems: 17
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in $S(a, b, c)$.
Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$:
\begin{align*} x^2+py+q&=0,\\
y^2+px+q&=0.
\end{align*}
Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.
The following sentece is written on a board:
[center]The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.[/center]
Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?
Show that the quadratic equation $x^2 + 7x - 14 (q^2 +1) =0$ , where $q$ is an integer, has no integer root.
Given a quadratic trinomial $f\left(x\right)=x^2+ax+b$. Assume that the equation $f\left(f\left(x\right)\right)=0$ has four different real solutions, and that the sum of two of these solutions is $-1$. Prove that $b\leq -\frac14$.
Solve:
$x^2(2-x)^2=1+2(1-x)^2$
Where $x$ is real number.
$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation
$$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.
Given a quadratic trinomial $f\left(x\right)=x^2+ax+b$. Assume that the equation $f\left(f\left(x\right)\right)=0$ has four different real solutions, and that the sum of two of these solutions is $-1$. Prove that $b\leq -\frac14$.
Determine all real numbers $a,b,c$ and $d$ with the following property: The numbers $a$ and $b$ are distinct roots of $2x^2-3cx+8d$ and the numbers $c$ and $d$ are distinct roots of $2x^2-3ax+8b$.
Let $p$ and $q$ be real numbers such that the quadratic equation $x^2 + px + q = 0$ has two distinct real solutions $x_1$ and $x_2$. Suppose $|x_1-x_2|=1$, $|p-q|=1$. Prove that $p, q, x_1, x_2$ are all integers.
Let $n$ be a composite positive integer. Let $1=d_1<d_2<\cdots<d_k=n$ be the positive divisors of $n.{}$ Assume that the equations $d_{i+2}x^2-2d_{i+1}x+d_i=0$ for $i=1,\ldots,k-2$ all have real solutions. Prove that $n=p^{k-1}$ for some prime number $p.{}$
Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$?
$ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$
$\textbf{(B)}\ \text{The discriminant is 9}\qquad$
$\textbf{(C)}\ \text{The roots are imaginary}\qquad$
$\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$
$\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $
Determine all pairs of real numbers $a$ and $b$, $b > 0$, such that the solutions to the two equations
$$x^2 + ax + a = b \qquad \text{and} \qquad x^2 + ax + a = -b$$
are four consecutive integers.
We consider the equation $x^2 + (a + b - 1)x + ab - a - b = 0$, where $a$ and $b$ are positive integers with $a \leq b$.
a) Show that the equation has $2$ distinct real solutions.
b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and $b < 2a.$
Determine all real numbers $a$ for which the system of equations
\begin{align*}
3x^2+2y^2+2z^2&=a\\
4x^2+4y^2+5z^2&=1-a
\end{align*}
has at least one solution $(x,y,z)$ in the real numbers.
If quadratic equations $x^2+ax+b=0$ and $x^2+px+q=0$ share one similar root then find quadratic equation for which has roots of other roots of both quadratic equations .
Determine all real number $(x,y)$ pairs that satisfy the equation. $$2x^2+y^2+7=2(x+1)(y+1)$$