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hey my lecturer put this example up for an exam tomorrow, could someone please explain how he gets to the 3rd line? is he using factorization?

$$V_{\mathrm{dsb}}=V_{\mathrm{gsb}}-V_{\mathrm t}$$

$$V_{\mathrm{dsb}}=V_{\mathrm{dd}}-(R_{\mathrm d}/2)K_n(V_{\mathrm{gsb}}-V_{\mathrm t})^2$$

$$V_{\mathrm{gsb}}=V_{\mathrm t}+(\sqrt{2K_{\mathrm n}R_{\mathrm d}V_{\mathrm{dd}}+1}-1)/K_{\mathrm n}R_{\mathrm d}$$

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I'm sorry, but your equation is unreadable right now. I'm guessing some of the letters are supposed to be subindices. Are the variables supposed to be $V_{dsb}$, $V_{dd}$, $R_d$, $K_n$, $V_{gsb}$, and $V_t$? – Arturo Magidin Sep 22 '11 at 13:16
I formatted the equations. It was rather non-trivial to guess what you meant; please check that everything is as it should be. You can right-click on the equations and select "Show Source" to see how to do the formatting so you can do it yourself next time. – joriki Sep 22 '11 at 13:28

The first equation states that the expression being squared in the second equation is $V_{\mathrm{dsb}}$. Thus the second equation becomes

$$V_{\mathrm{dsb}}=V_{\mathrm{dd}}-(R_{\mathrm d}/2)K_nV\;_{\mathrm{dsb}}^2\;.$$

This is a quadratic equation for $V\;_{\mathrm{dsb}}$. Substituting one of its two solutions into the first equation yields the third equation.

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sorry about the formatting, thanks for the help it just wasn't clicking, and thanks for the formatting help – michael Sep 22 '11 at 14:05

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