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If we have $f(x)=(2A(A+\delta)-1)x^{2}-2Ax+1$, then the values of $x$ are equal to $\frac{1}{A+\sqrt{1-A(2\delta+A)}}$ and $\frac{1}{A-\sqrt{1-A(2\delta+A)}}$. The question is how to find the solutions (either negative or positive value) of $f(z)$ when $2A(A+\delta) <1 $ and $2A(A+\delta) >1 $ where $A= \alpha \cos(\delta)$

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Your $f$ is a quadratic function; the quadratic formula remains applicable with symbolic coefficients... –  J. M. Oct 17 '11 at 15:13
    
@J.M.: I am sorry, A is actually $\alpha \cos{\delta}$ –  DRN Oct 17 '11 at 15:27
    
That doesn't prevent you from solving for $x$... –  J. M. Oct 17 '11 at 15:28
    
@J.M., actually this is one of the step to find covering theorem for certain convex function. I have to solve this in order to make sure the value of x fulfilled that convex function. –  DRN Oct 17 '11 at 15:35

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