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authorHolden Rohrer <hr@hrhr.dev>2020-04-14 12:18:22 -0400
committerHolden Rohrer <hr@hrhr.dev>2020-04-14 12:18:22 -0400
commitf367c3918b1f068db4032a67bd489315719f2e9e (patch)
tree5020082315a787e0f08d7936ec7b8991136706b5 /execsumm
parent397cf672a4ac67ceae1f045856dfa35ffee9f851 (diff)
added quadratic formula solution
Diffstat (limited to 'execsumm')
-rw-r--r--execsumm/document.tex10
1 files changed, 6 insertions, 4 deletions
diff --git a/execsumm/document.tex b/execsumm/document.tex
index 32d4a0a..c308d05 100644
--- a/execsumm/document.tex
+++ b/execsumm/document.tex
@@ -22,12 +22,14 @@ $$\x' =
$$
The characteristic polynomial is
$$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1\rload)
-+ C_2^2R_1\rload).$$
++ C_2C_1R_1\rload).$$
Expanded,
$$-\lambda( \lambda^2 + \lambda(C_2(R_1+\rload)+C_1\rload)
-+ C_1C_2\rload(R_1+\rload) + C_2^2R_1\rload).$$
-In terms of its roots,
-$$???$$
++ C_1C_2\rload(2R_1+\rload)).$$
+In terms of its roots (with $b=C_2(R_1+\rload)+C_1\rload$ and
+$c = C_1C_2\rload(2R_1+\rload),$
+$$-\lambda{(\lambda-{-b+\sqrt{b^2-4c}\over2})
+(\lambda-{-b-\sqrt{b^2-4c}\over 2})}$$
\def\bu{\par\leavevmode\llap{\hbox to \parindent{\hfil $\bullet$ \hfil}}}
\opt{.15fil}