Intersection of a line and a parabola

Text

Find the intersection points of the parabola $y=x^{2}-3$ and the line $y=2x$.

Solution

Set the two equations equal by matching the $y$-values: $x^{2}-3=2x\;\Rightarrow\; x^{2}-2x-3=0\;\Rightarrow\;(x+1)(x-3)=0,$ so $x=-1$ or $x=3$. The corresponding ordinates (from $y=2x$) are $y=-2$ and $y=6$. The intersection points are $(-1,-2)$ and $(3,6)$. $\boxed{\,(-1,-2)\ \text{and}\ (3,6)\,}$