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# help

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Solve the system x = y^2 + y + 1, 5y = 2 - x - x^2.

Nov 16, 2019

#1
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Here is a Desmos graphical solution, which might help to find the algebraic solution....which I think might be kind of messy,,,,,,

Nov 16, 2019
#2
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Solve the system x = y^2 + y + 1, 5y = 2 - x - x^2.

Nov 16, 2019
edited by Omi67  Nov 16, 2019
#3
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x = y^2 + y + 1         (1)

5y  = 2 - x - x^2      (2)

Square both sides of (1)

x^2  =  (y^2 + y + 1) (y^2 + y + 1)

x^2  =  y^4 + y^3 + y^2 + y^3 + y^2 + y  + y^2 + y + 1

x^2  =  y^4 + 2y^3 + 3y^2  + 2y+  1      (3)

Sub (1) and (3)  into (2)  for x^2 and x

5y  = 2 - (y^2 + y + 1) - ( y^4 + 2y^3 + 3y^2  +2y +  1)

5y  =  2  - y^2 - y - 1 - y^4 - 2y^3 - 3y^2 - 2y - 1

5y  = -y^4 - 2y^3 - 4y^2 -3 y

y^4 + 2y^3 + 4y^2 + 8y   =  0      factor

y^3 ( y + 2)  + 4y ( y + 2)  = 0

(y^3 + 4y)  ( y + 2)  = 0

y ( y^2 + 4) ( y + 2)  = 0

The second factor does not provide real solutions

The  other two solutions are

y  = 0       and       y + 2  = 0

y  = -2

When y  = 0     x = (0)^2 + 0 + 1  =   1

When y  = -2   x  = (-2)^2  - 2 + 1   = 3

So....the solutions are

(1, 0)   and  ( 3, -2)    .....as EP found graphically  !!!!

Nov 16, 2019