Quadratic Equat...

Given $$\alpha $$ and $$\beta $$ are the roots of the equation $${x^2} - 4x + k = 0(k \ne 0)$$. If $$\alpha \beta ,\ \alpha {\beta ^2} + {\alpha ^2}\beta ,\ {\alpha ^3} + {\beta ^3}$$ are in geometric progression then the value of $$'k'$$ equals

The value of $$\sqrt {6 + \sqrt {6 + \sqrt {6 +  \ldots  \ldots  \ldots  \ldots \infty } } } $$ is 

The equation $$ 2 sin^2 x-(p+3) sin x+2p-2=0 $$ posses a real solution, if :

Solve by the method of completing the square $$5x^2 - 6x - 2 = 0$$

Let $$f\left( x \right) ={ x }^{ 3 }+a{ x }^{ 2 }+bx+5\sin ^{ 2 }{ x } $$ be an increasing function in the in the set of real numers $$R$$.Then $$a$$ and $$b$$ satisfy the condition.

If $$\alpha $$ and $$\beta $$ are the roots of equation $${x^2} - 3x + 1 = 0$$ and $${a_n} = {\alpha ^n} + {\beta ^n},n \in N$$ then the value of $$\dfrac{{{a_7} + {a_5}}}{{{a_6}}}$$

The sum of the roots of the quadratic $$5x^{2}-6x+1=0$$ is

If one root of $$2x^{2}+kx+1=0$$ is $$\dfrac{1}{2}$$, then the value of $$k$$ is ____.

If $$\alpha,\beta$$ are the roots of $${x}^{2}-x+1=0$$ then $$\cfrac { { \left( { \alpha  }^{ 2 }-\alpha  \right)  }^{ 3 }+{ \left( { \beta  }^{ 2 }-\beta  \right)  }^{ 3 } }{ \left( 2-\alpha  \right) \left( 2-\beta  \right)  } $$ is equal to

The roots of  $$a{x^2} + bx + c = 0$$ where $$a \ne 0$$ and coefficients are real and $$a+c <b$$, then