7 Geometrical representation of quadratics

We can plot a graph of the function $y=a{x}^2+bx+c$ (given the values of $a,b$ and $c$ ). If the graph crosses the horizontal axis it will do so when $y=0$ , and so the $x$ coordinates at such points are solutions of $a{x}^2+bx+c=0$ . Depending on the sign of $a$ and of the nature of the solutions there are essentially six different types of graph that can occur. These are displayed in Figure 4.

Figure 4 :

Sometimes a graph of the quadratic is used to locate the solutions; however, this approach is generally inaccurate. This is illustrated in the following example.

Example 22

Solve the equation $x^2-4x+1=0$ by plotting a graph of the function:

$y=x^2-4x+1$

Solution

By constructing a table of function values we can plot the graph as shown in Figure 5.

Figure 5 :

The solutions of the equation $x^2-4x+1=0$ are found by looking for points where the graph crosses the horizontal axis. The two points are approximately $x=0.3$ and $x=3.7$ marked C and D on the Figure.

Exercises

1. Solve the following quadratic equations giving exact numeric solutions. Use whichever method you prefer

   (a) $x^2-9=0$	(b) $s^2-25=0$
   (c) $3x^2-12=0$	(d) $x^2-5x+6=0$
   (e) $6s^2+s-15=0$	(f) $p^2+7p=0$

2. Solve the equation $2x^2-3x-7=0$ giving solutions rounded to 4 d.p.
3. Solve the equation $2t^2+3t-4$ giving the solutions in surd form.

Answer