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.