How do you solve simultaneous equations where one is quadratic?

A set of two independent linear equations with two unknowns has one solution pair.

If one of the equations is quadratic, there will be two solution pairs.

A quadratic equation is solved by finding two solutions, for example x² = 9 has solutions 3 and -3. If there is a y unknown involved and another equation given, that means that you first need to figure out what the y is. Once you figure it out, you substitute the value (or expression) of y into your quadratic equation, and solve it as you would any other quadratic equation: by getting two solutions.

[Note that y can also have an exponent ², not necessarily x. In this case, first solve for x, and substitute its value or expression into the quadratic equation.]

Example:

x²+2y+1 = 0

y+3 = -2x

y = -2x-3

x²+2(-2x-3)+1 = 0

x²-4x-6+1 = 0

x²-4x-5 = 0

x₁ = -1, x₂ = 5

y₁ = -2(-1)-3 = 2-3 = -1

y₂ = -2*5-3 = -10-3 = -13

Solutions: (-1, -1), (5, -13)