Descartess Method of Equal Roots Descartess method for

Chapter 12, Problem 91

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Descartess Method of Equal Roots Descartess method for finding tangents depends on the idea that, for many graphs, the tangent line at a given point is the unique line that intersects the graph at that point only. We will apply his method to find an equation of the tangent line to the parabola y = x at the point 12, 42. See the figure. First, we know that the equation of the tangent line must bein the form Using the fact that the pointis on the line, we can solve for b in terms of m and get theequation Now we want to bethe unique solution to the systemby = x2y = mx + 4 - 2m From this system, we get Byusing the quadratic formula, we getTo obtain a unique solution for x, the two roots must beequal; in other words, the discriminantmust be 0. Complete the work to get m, and write anequation of the tangent line.