A Circle and a Line

The same method works for curves that are not parabolas. Take x^2 + y^2 = 25. By Pythagoras, a point (x, y) is at distance 5 from the origin exactly when this equation holds, so its graph is the circle of radius 5 centred at (0, 0). The number on the right is the radius squared.

To meet it with the line y = x + 1, substitute the line into the circle: x^2 + (x + 1)^2 = 25. Expand the bracket, x^2 + x^2 + 2x + 1 = 25, then collect and move the 25: 2x^2 + 2x - 24 = 0. Divide by 2 and factor: (x + 4)(x - 3) = 0, so x is -4 or 3.

Find each y from the line: x = 3 gives y = 4, and x = -4 gives y = -3. The meeting points are (3, 4) and (-4, -3), and both check in the circle, since 9 + 16 and 16 + 9 are 25. Substitute into the circle, not the other way round: only the line says what y is.