Answer
$(1/2, \sqrt {7/2})$
$(1/2, - \sqrt {7/2})$
$(-1, \sqrt 2)$
$(-1, -\sqrt 2)$
Work Step by Step
Given the system of equations:
1. $x - y^2 + 3 = 0$
2. $2x^2 + y^2 - 4 = 0$
It asks to find all solutions.
Add equations one and two
$x - y^2 + 3 = 0$
+($2x^2 + y^2 - 4 = 0$)
$2x^2 + x -1 = 0$
$(2x - 1) (x + 1) = 0$
$x = 1/2, -1$
Substitute the values into the equation
$1/2 - y^2 + 3 = 0$
$y^2 = 7/2$
$y = +/- \sqrt {7/2}$
$-1 - y^2 + 3 = 0$
$2 = y^2$
$y = +/- \sqrt 2$
All solutions:
$(1/2, \sqrt {7/2})$
$(1/2, - \sqrt {7/2})$
$(-1, \sqrt 2)$
$(-1, -\sqrt 2)$
