In this exercise we investigate the effect of the constant c upon the roots of the quadratic equation x2 − 6x + c = 0. We do this by looking at the x-intercepts of the graphs of the corresponding equations y = x2 − 6x + c.
(a) Set a viewing rectangle that extends from 0 to 5 in the x-direction and from −2 to 3 in the y-direction. Then (on the same set of axes) graph the equations y = x2 − 6x + c with c running from 8 to 10 at increments of 0.25. In other words, graph the equations y = x2 − 6x + 8, y = x2 − 6x + 8.25, y = x2 − 6x + 8.50, and so on, up through y = x2 − 6x + 10.
(b) Note from the graphs in part (a) that, initially, as c increases, the x-intercepts draw closer and closer together. For which value of c do the two x-intercepts seem to merge into one?
(c) Use algebra as follows to check your observation in part (b). Using that value of c for which there appears to be only one intercept, solve the quadratic equation x2 − 6x + c = 0. How many roots do you obtain?
(d) Some of the graphs in part (a) have no x-intercepts. What are the corresponding values of c in these cases? Pick any one of these values of c and use the quadratic formula to solve the equation x2 − 6x + c = 0. What happens?
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