Many equations can be expressed in the form \[\begin{equation*}f(x) = \phi(x), \tag{1}\end{equation*}\] where \(f(x)\) and \(\phi(x)\) are functions whose graphs are easy to draw. And if the curves \[y = f(x),\quad y = \phi(x)\] intersect in a point \(P\) whose abscissa is \(\xi\), then \(\xi\) is a root of the equation (1).

Examples XVII

1. The quadratic equation. \(ax^2 + 2bx + c = 0.\)

This may be solved graphically in a variety of ways. For instance we may draw the graphs of \[y = ax + 2b,\quad y = -c/x,\] whose intersections, if any, give the roots. Or we may take \[y = x^{2},\quad y = -(2bx + c)/a.\] But the most elementary method is probably to draw the circle \[a(x^{2} + y^{2}) + 2bx + c = 0,\] whose centre is \((-b/a, 0)\) and radius \(\{\sqrt{b^{2} – ac}\}/a\). The abscissae of its intersections with the axis of \(x\) are the roots of the equation.

2. Solve by any of these methods \[x^{2} + 2x – 3 = 0,\quad x^{2} – 7x + 4 = 0,\quad 3x^{2} + 2x – 2 = 0.\]

3. The equation \(x^m + ax + b = 0\). This may be solved by constructing the curves \(y = x^{m}\), \(y = -ax – b\). Verify the following table for the number of roots of \[\begin{gathered} x^{m} + ax + b = 0: \\ \begin{alignedat}{3} &(a) &&m~\text{even} &&\left\{ \begin{aligned} &\text{$b$ positive, two or none,}\\ &\text{$b$ negative, two;} \end{aligned} \right. \\ &(b) &&m~\text{odd} &&\left\{ \begin{aligned} &\text{$a$ positive, one,}\\ &\text{$a$ negative, three or one.} \end{aligned} \right. \end{alignedat}\end{gathered}\] Construct numerical examples to illustrate all possible cases.

4. Show that the equation \(\tan x = ax + b\) has always an infinite number of roots.

5. Determine the number of roots of \[\sin x = x,\quad \sin x = \tfrac{1}{3} x,\quad \sin x = \tfrac{1}{8} x,\quad \sin x = \tfrac{1}{120} x.\]

6. Show that if \(a\) is small and positive ( \(a = .01\)), the equation \[x – a = \tfrac{1}{2}\pi\sin^{2} x\] has three roots. Consider also the case in which \(a\) is small and negative. Explain how the number of roots varies as \(a\) varies.

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