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How many solutions (2D)?

Linear equations (straight lines) can do 3 things:

  1. intersect each other (one solution, which is where the x,y value of the intersection satisfies both equations)

  2. be parallel (no solutions), which means they have the same slope

  3. be on top of each other, in which case they are the same line, and have the same x, y values at all points, giving an infinite number of solutions.

Quadratic equations (highest power of x = 2), parabolas, which intersect either a line or another parabola.

  1. You are looking for the point in space at which they share the same value of x and y. Either set the two equations equal to each other, or substitute the variable that has a maximum degree of 1.

  2. Rearrange the remaining variable into an equation in the form ax^2 + bx + c = 0

  3. Calculate the discriminant i.e. D = b^2 - 4ac

  4. If D > 0 there are 2 solutions to the equation in (2); if D = 0 there is just one, and if D < 0 there are no real solutions. Why? Because we have an algebraic solution to (2) which takes the form x = -(b/2a) +- (sqrt(D)/2a) If D > 0 there are two solutions equidistant from -(b/2a); if D = 0 then there is only the solution x = -(b/2a); if D < 0 then there are no real roots to the square root, or to x.

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