Question 0.1.23: Finding the Intersections of a Line and a Parabola Find th...
Finding the Intersections of a Line and a Parabola
Find the points of intersection of the parabola y = x² − x − 5 and the line y = x + 3.
A sketch of the two curves (see Figure 0.25 on the following page) shows that there are two intersections, one near x = −2 and the other near x = 4. To determine these precisely, we set the two functions equal and solve for x:
x² − x − 5 = x + 3.
Subtracting (x + 3) from both sides leaves us with
0 = x² − 2x − 8 = (x − 4)(x + 2).
This says that the solutions are exactly x = −2 and x = 4. We compute the corresponding y-values from the equation of the line y = x + 3 (or the equation of the parabola). The points of intersection are then (−2, 1) and (4, 7). Notice that these are consistent with the intersections seen in Figure 0.25.
