Mathematics Asked by Karagum on December 28, 2020

There is a given system of equations

begin{align} ab+c+d&=phantom{-}4 \ ad+bc&=-8 \ a+b&=phantom{-}5 \ cd&=-8 end{align}

I have tried to simplify it by multiplying one equation by another or adding one to another or similar. Nothing came up.

Any hints would be great! Thank you.

From the 3rd and 4th equations we find that $$b=5-aquad text{ and }quad d=-8/c. tag{1}$$ Substituting these in the 1st and 2nd equations they become $$ x:=(c^2+5ac-a^2c-4c-8)/c tag{2} = 0$$ and $$ y:=(5c^2-ac^2-8a+8c)/c = 0. tag{3}$$ The polynomial resultant of $,xcdot c,$ and $,ycdot c,$ eliminating $,c,$ is $$ 8(a-2)(a-3)(a^2-4a-1)(a^2-6a+4). tag{4}$$ This has six roots for $,a$. For each value of $,a,$ the GCD of $,x,$ and $,y,$ uniquely determines $,c,$ and the values for $,b,$ and $,d,$ are uniquely determined from equations in $(1)$.

Of course, there is nothing unique about picking the 3rd and 4th equations and solving for $b$ and $d$. Also, I decided to use resultants to solve for $a$, but there are alternative ways to do so.

Correct answer by Somos on December 28, 2020

begin{align} ab+c+d&=-4 tag{1}label{1} ,\ ad+bc&=-8 tag{2}label{2},\ a+b&=-5 tag{3}label{3},\ cd&=-8 tag{4}label{4}.end{align}

Substitution of

begin{align} a&=5-b tag{5}label{5} ,\ d&=-frac8c tag{6}label{6} end{align}

into eqref{2} results in

begin{align} b&=frac{40-8c}{c^2+8} tag{7}label{7} ,\ a &= frac{c(5c+8)}{c^2+8} tag{8}label{8} . end{align}

Next, substitution of eqref{6}-eqref{8} into eqref{1} gives an equation in $c$

begin{align} frac{136c^3-32c^4+256c^2+c^6-512}{c(c^2+8)^2} &= 4 tag{9}label{9} , end{align}

which is equivalent to

begin{align} (c-2)(c+4)(c^2-4c-16)(c^2-2c-4)&=0 tag{10}label{10} end{align}

with six real roots

begin{align} { 2,-4, 1+sqrt5, 1-sqrt5, 2+2sqrt5, 2-2sqrt5 } tag{11}label{11} . end{align}

Expressions eqref{6}-eqref{8} provide corresponding values of $d,b$ and $a$.

Answered by g.kov on December 28, 2020

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- haakon.io on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP