Mathematics Asked by YoYo12 on December 17, 2020
Is there an algebraic method to solve the following system in $ mathbb{C} $,
$$ begin{cases} 2x^2 +x + z = 0 \ (1-z) x^2 + 3x – y = 0 \ (2+y) x^2 + 4x – 2 = 0 end{cases} $$ ?.
Here, $ y not in { 0, -2 } $, and $ z not in { 0,1 } $.
Thanks in advance for your help.
$$ begin{cases} 2x^2 +x + z = 0 \ (1-z) x^2 + 3x - y = 0 \ (2+y) x^2 + 4x - 2 = 0 end{cases} $$
subtract equation 1 from 3 :
$$yx^2+3x-z=0$$
Make a system with equation 2:
$$ begin{cases} yx^2+3x-z=0 \ (1-z) x^2 + 3x - y = 0 end{cases} $$
ُThe value of x from both equations must be equal, so we must have:
$x=frac {3 ± sqrt{9+4yz}}y=frac{3 ±sqrt{9+4y(1-z)}}{1-z}$
Now we may assume that:
$3 ±sqrt{9+4yz}= 3 ±sqrt{9+4y(1-z)}$ ⇒ $yz=y(1-z)$
$y=1-z$
Now solve following system and find y and z. Plugging z in first equation will give you two roots for x.
$$ begin{cases} y(1-z)=yz\ y=1-z end{cases} $$
Answered by sirous on December 17, 2020
Hijacking the concept behind Michael Rozenberg's comment, the first thing to do is check that you copied the problem correctly.
Assuming so,
let $f(x) = -(2x^2 + x)$
and let $g(x) = [1 - f(x)]x^2 + 3x.$
From the first equation, $z = f(x)$, so
the 2nd equation gives $y = g(x).$
Thus, the 3rd equation becomes
$[2 + g(x)]x^2 + 4x - 2 = 0.$
Answered by user2661923 on December 17, 2020
Get help from others!
Recent Answers
Recent Questions
© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP