All Questions
9 questions
4
votes
3
answers
119
views
Find the condition for the three equations $a_rx^2+b_rx+c_r=0$; $r=1,2,3$ to have a common root.
Find the condition for the three equations $a_rx^2+b_rx+c_r=0$; $r=1,2,3$ to have a common root.
My attempt is as follows:
\begin{equation}
a_1x^2+b_1x+c_1=0\tag{1}
\end{equation}
\begin{equation}
...
- 3,500
1
vote
2
answers
139
views
If each pair of equations $x^2=b_1x+c_1=0,x^2=b_2x+c_2 \text{ and } x^2+b_3x=c_3$ have a common root, prove following
If each pair of equations $x^2=b_1x+c_1=0,x^2=b_2x+c_2 \text{ and } x^2+b_3x=c_3$ have a common root, prove that
$(b_1+b_2+b_3)^2=4(c_1+c_2+c_3+b_1b_2)$
My attempt is as follows:
For equations $x^2=...
- 3,500
1
vote
2
answers
116
views
If $a^2+c^2>ab$ and $b^2>4c^2$ , for real x, show that $\frac{x+a}{x^2+bx+c^2}$ cannot lie between two limits
If $a^2+c^2>ab$ and $b^2>4c^2$ , for real x, show that $\frac{x+a}{x^2+bx+c^2}$ cannot lie between two limits
My attempt is as follows:
$$y=\frac{x+a}{x^2+bx+c^2}$$
$$yx^2+byx+yc^2=x+a$$
$$yx^...
- 3,500
2
votes
3
answers
139
views
$\{a^3+(1-\sqrt{2})a^2-(3+\sqrt{2})a+3\sqrt{2}\}x^2+2(a^2-2)x+a>-\sqrt{2}$
If $\{a^3+(1-\sqrt{2})a^2-(3+\sqrt{2})a+3\sqrt{2}\}x^2+2(a^2-2)x+a>-\sqrt{2}$ is satisfied for all real $x>0$ then obtain the possible values of the parameter $a$.
My attempt is as follows:
$$\...
- 3,500
3
votes
1
answer
80
views
Find exhaustive range of $k$ such that $f(x)=\frac{x-1}{k-x^2}$ never belongs to $\left[-1 \:\: \frac{-1}{3}\right]$
Find exhaustive range of $k$ such that $$f(x)=\frac{x-1}{k-x^2}$$ never belongs to $\left[-1 \:\: \frac{-1}{3}\right]$
My try:
Letting $$y=\frac{x-1}{k-x^2}$$ we get
$$yx^2+x-(1+ky)=0$$ and since $...
- 13.3k
2
votes
4
answers
3k
views
Quadratic Equation with imaginary roots.
I know that if the discriminant of a quadratic equation is less than $0$, the roots are imaginary.
But why is this quadratic expression (with imaginary roots) always positive for all values of $x$?
...
- 4,993
-3
votes
1
answer
52
views
For which values of $m$ we get: $x_1>4$ and $x_2<1$? [closed]
given this equation:
$$x-\sqrt{x}(3+m)-2(1-m^2)=0$$
with roots $x_1$,$x_2$.
For which values of $m$ we get: $x_1>4$ and $x_2<1$?
- 419
0
votes
1
answer
65
views
Solving quadratic inequalities $a^2-b^2+2b-1\geq0$....
Let $a,b,p,q,r\in\mathbb{R}$ such that $p,q>0$. I need to find or guess some $p,q,r$ such that
\begin{gather}
a^2-b^2+2b-1\geq0\\
\Downarrow\\
a^2\left[4(r-p)q-r^2\right]-b^2[4p^2-4pq]+2b\left(r(...
- 899
1
vote
2
answers
68
views
Cyclic inequality, need help [closed]
$x+\frac{1}{y}=10$;
$y+\frac{1}{z}=10$;
$z+\frac{1}{x}=10$;
What is the highest possible value of z?
- 215