Answer
$(0, 0)$
$(1, -1)$
$(-2, -4)$
Work Step by Step
Given the system of equations:
1. $x^2 + y = 0$
2. $x^3 - 2x - y = 0$
It asks to find all solutions.
Add equation one and equation two
$x^2 + y = 0$
+ ($x^3 - 2x - y = 0$)
$x^3 + x^2 - 2x = 0$
$x(x^2 + x - 2) = 0$
$x(x+2)(x-1) = 0$
$x = 0, 1, -2$
Substitute the values into the equation
$0^2 + y = 0$
$y = 0$
$1^2 + y = 0$
$y = -1 $
$(-2)^2 + y = 0$
$y = -4$
All solutions:
$(0, 0)$
$(1, -1)$
$(-2, -4)$
