30.Graphical solution of equations

1. The quadratic equation. $a{x}^2+2bx+c=0.$

This may be solved graphically in a variety of ways. For instance we may draw the graphs of $$y=ax+2b,y=-c/x,$$ whose intersections, if any, give the roots. Or we may take $$y=x^2,y=-(2bx+c)/a.$$ 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 $$x^2+2x–3=0,x^2–7x+4=0,3x^2+2x–2=0.$$

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 $$\begin{array}{c}{x}^m+ax+b=0:\\ \begin{array}{rlrlrl}& (a)& & m\text{even}& & \{\begin{array}{rl}& b\text{ positive, two or none,}\\ & b\text{ negative, two;}\end{array}\\ & (b)& & m\text{odd}& & \{\begin{array}{rl}& a\text{ positive, one,}\\ & a\text{ negative, three or one.}\end{array}\end{array}\end{array}$$ 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 $$\sin x=x,\sin x=\frac{1}{3}x,\sin x=\frac{1}{8}x,\sin x=\frac{1}{120}x.$$

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