The solutions are [tex](\sqrt{2},-1) \text { and }(-\sqrt{2},-1)[/tex].
Solution:
Given equation:
[tex]-2 x^{2}+y=-5[/tex]
Add [tex]2 x^{2}[/tex] on both sides.
[tex]-2 x^{2}+y+2 x^{2}=-5+2 x^{2}[/tex]
[tex]y=-5+2 x^{2}[/tex] -------- (1)
[tex]y=-3 x^{2}+5[/tex] (given) -------- (2)
Equate (1) and (2).
[tex]-5+2 x^{2}=-3 x^{2}+5[/tex]
Add [tex]3 x^{2}[/tex] on both sides.
[tex]-5+2 x^{2}+3 x^{2}=-3 x^{2}+5 +3 x^{2}[/tex]
[tex]-5+5x^{2}=5[/tex]
Add 5 on both sides.
[tex]-5+5x^{2}+5=5+5[/tex]
[tex]5x^{2}=10[/tex]
Divide by 5 on both sides, we get
[tex]x^{2}=2[/tex]
Taking square root on both sides, we get
[tex]x=\sqrt{2}, x=-\sqrt{2}[/tex]
Substitute [tex]x=\sqrt{2}[/tex] in (1).
[tex]y=-5+2(\sqrt{2})^2[/tex]
[tex]y=-5+4[/tex]
[tex]y=-1[/tex]
Substitute [tex]x=-\sqrt{2}[/tex] in (1).
[tex]y=-5+2(-\sqrt{2})^2[/tex]
[tex]y=-5+4[/tex]
[tex]y=-1[/tex]
Therefore the solutions are [tex](\sqrt{2},-1) \text { and }(-\sqrt{2},-1)[/tex].
Option C is the correct answer.
