Imaginary Roots and the Graph

Graphical illustration of imaginary roots

The solution of x^2 + 1 = 0 gives x = \pm\sqrt{-1}. These roots are imaginary numbers. Just now it is desirable to know that many quadratic equations have imaginary roots. It is instructive to see what happens when we try to solve such an equation graphically. Consider the three equations

• I. x^2 - 4x - 5 = 0,
• II. x^2 - 4x + 4 = 0,
• III. x^2 - 4x + 13 = 0.

The graphs of the functions in their left members are given in the figure. The three functions differ only in their constant terms: 9 added to the constant term of I gives that of II, and 9 added to the constant term of II gives that of III. As the constant term increases, the graph rises. It does not change its shape, nor does it move to the left or the right.

From the graph, the roots of x^2 - 4x - 5 = 0 are 5 and -1. Factoring gives the same: (x - 5)(x + 1) = 0.

If curve I moves upward, its two roots change in value and become the single root of curve II, which touches the x-axis at the point where x equals 2. Factoring x^2 - 4x + 4 = 0 gives (x - 2)(x - 2) = 0, so x = 2.

If curve I moves still farther upward, to the position of III, it no longer meets the x-axis, and we cannot expect the graph to show roots of x^2 - 4x + 13 = 0. It does show that the value of x^2 - 4x + 13 at the lowest point of the curve is 9. This means that for every real value of x, positive or negative, x^2 - 4x + 13 is never less than 9. No real number, substituted for x, makes it zero. In the next chapter an algebraic method of solving any quadratic is given; that method will show the roots of x^2 - 4x + 13 = 0 to be the imaginary expressions 2 + 3\sqrt{-1} and 2 - 3\sqrt{-1}.

y = x^2 - 4x + 13

The steps

The real roots of a quadratic equation can be obtained by these steps:

1. Transpose the terms so that the second member is zero.
2. Graph the function in the first member.
3. Note the x-distance of each point at which the curve crosses (or touches) the x-axis. These x-distances are the roots of the quadratic.

If the graph does not cross or touch the x-axis, the roots are not real. You may still regard such an equation as having roots, and call them imaginary. For these, type none (no real root).

Check your graph with the calculator. Open the calculator's Graph tab and type the function with y in place of f(x), for example y = 2x + 3, or press the Graph button beside an equation in the lesson. Zoom in where the curve meets the x-axis to read a root more closely. Draw the graph on paper first, though: the point is to learn to read it yourself.

Try one: