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: ${x}^{2}+{\left(x+1\right)}^{2} = 25$. Expand the bracket, ${x}^{2}+{x}^{2}+2x+1 = 25$, then collect and move the 25: $2{x}^{2}+2x-24 = 0$. Divide by 2 and factor: $\left(x+4\right)\left(x-3\right) = 0$, 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.