Show that t = 1/2 solves the equation

[hau.sim]

Homework Statement

Show that t-hat = 1/2 solves the equation below.

Now, I know I can just plug in 1/2, where t-hat is, and see that both sides are equal to zero, but is it possible to isolate t-hat. Say, if wanted to find t, and did not know the value already..?

$$\[2\hat t - 1 = \frac{{\sigma _F^{}}}{{2\sigma _z^{}}}\left( {\frac{1}{{\sqrt {1 - \hat t} }} - \frac{1}{{\sqrt {\hat t} }}} \right)\]$$

Homework Equations

$$\[\hat t \in [0,1]\]$$

$$\[\begin{array}{l}<br /> z \sim N\left( {0,\sigma _z^2} \right)\\<br /> F \sim N\left( {\bar F,\sigma _F^2} \right)<br /> \end{array}\]$$

The Attempt at a Solution

$$\[\begin{array}{l}<br /> 2\hat t - 1 = \frac{{\sigma _F^{}}}{{2\sigma _z^{}}}\left( {\frac{1}{{\sqrt {1 - \hat t} }} - \frac{1}{{\sqrt {\hat t} }}} \right)\\<br /> 4{{\hat t}^2} - 4\hat t + 1 = \frac{{\sigma _F^2}}{{4\sigma _z^2}}{\left( {\frac{1}{{\sqrt {1 - \hat t} }} - \frac{1}{{\sqrt {\hat t} }}} \right)^2}\\<br /> 4{{\hat t}^2} - 4\hat t + 1 = \frac{{\sigma _F^2}}{{4\sigma _z^2}}\left[ {\frac{1}{{1 - \hat t}} + \frac{1}{{\hat t}} - 2\frac{1}{{\sqrt {1 - \hat t} \sqrt {\hat t} }}} \right]<br /> \end{array}\]$$

Thx, any help is appreciated, P DK

 

[Berko]

From your last line, isolate the square root term and square both side...get rid of those square root signs.

 

[hau.sim]

Hi Berko

Thanks for your reply

I have tried the following, but I have no clue whether it is getting me in the right direction.

..

 

[Berko]

Your seventh line of work...you squared the right side but not the left.

 

[Berko]

Your ninth line of work...one of your minus signs should be a plus sign.

Fix those and redo. I'll look at what you come up with.

 

[hau.sim]

Hi Berko

Thanks for your help. I never solved it, and the deadline was today... I think I got pretty close to a solution, but I don't know for sure...