Equations as graphical intersections

At an intersection, two graphs have the same output for the same input. Therefore the \(x\)-coordinates of their intersections solve the equation obtained by equating their expressions. To use a supplied curve, rearrange the equation so that the curve’s expression stands alone on one side.

Example

Using the curve \(y=x^2\), solve \(x^2-x-2=0\) by drawing \(y=x+2\). The intersections are \((-1,1)\) and \((2,4)\), giving \(x=-1\) and \(x=2\).

To solve \(x^2=3\) on the same curve, instead draw the horizontal line \(y=3\); the readings are about \(-1.73\) and \(1.73\). An equation asks for input values, not necessarily the complete intersection coordinates.