How to solve simultaneous equations with a quadratic

Example.

" Solve the simultaneous equations

y + 4x + 1 = 0

y^2 + 5x^2 + 2x = 0 "

We can rearrange the first equation to something that we can substitute into the second equation, for example

y = -4x - 1

We then substitute this into the second equation

(-4x - 1)^2 + 5x^2 + 2x = 0

16x^2 + 8x + 1 + 5x^2 + 2x = 0

21x^2 + 10x + 1 = 0

(7x + 1)(3x + 1) = 0

Gives us the solutions

x = -1/7 and x = -1/3

We then put these values into our first equation (y = -4x - 1) to give

y = -3/7 and y = 1/3