Tag: sign of quadratic expression

The number of real solution of $x-\dfrac{1}{x^2-4}=2-\dfrac{1}{x^2-4}$ is 

Let $f(x)=1+2x+3x^2+.....+(n+1)x^n,$ where n is even. Then the number of real roots of the equation $f(x)=0$ is 

The general solution of the equation 
$tan \, x + tan \, 2x + \sqrt{3} \, tan \, x \, tan \, 2x = \sqrt{3}$ is 

If $1+\surd {3}i/2$ is a root of equation $x^{4}-x^{3}+x1=0$ then its real roots are 

The equation
$\left| {\begin{array}{{20}{c}}  {{{\left( {1 + x} \right)}^2}}&{{{\left( {1 - x} \right)}^2}}&{ - \left( {2 + {x^2}} \right)} \   {2x + 1}&{3x}&{1 - 5x} \   {x + 1}&{2x}&{2 - 3x} \end{array}} \right| + \left| {\begin{array}{{20}{c}}  {{{\left( {1 + x} \right)}^2}}&{2x + 1}&{x + 1} \   {{{\left( {1 - x} \right)}^2}}&{3x}&{2x} \   {1 - 2x}&{3x - 2}&{2x - 3} \end{array}} \right| = 0$

If a,b,c and d are the real roots of the equation : $x^{4}+p _{1}x^{1}+p _{2}x^{2}+p _{3}x+p _{4}=0$ and $(1+a^{2})(1+b^{2})(1+c^{2})(1+d^{2})=k(1-p _{2}+p _{4})^{2}+(p _{3}-p _{1})^{2}$ then the value f k is:

Number of solutions of the equation $\cos 6x+\tan^2x+\cos 6x\tan^2x=1$ in the interval $[0, 2\pi]$ is?

At how many maximum points will a cubic equation cut the $x$ axis?

Find the point of intersection of $x+y=-1\x-y=15$

If $\alpha $ and $\beta $ are the roots of ${ x }^{ 2 }+px+q=0$ and ${ \alpha  }^{ 4 } , { \beta  }^{ 4 }$ are the roots of ${ x }^{ 2 }-rx+s=0$, then the equation ${ x }^{ 2 }-4qx+2{ q }^{ 2 }-r=0$ has always two real roots.