12.1. Solving a Quadratic Equation Graphically and by Calculator

Recall that a polynomial equation is one in which all powers of x are positive integers. A quadratic equation is a polynomial equation of second degree. That is, the highest power of x in the equation is 2. It is common practice to refer to a quadratic equation simply as a quadratic.

Example 1:

The following equations are quadratic equations:

  1. 4x2 − 5x + 2 = 0

  2. x2 = 58

  3. 9x2 − 5x = 0

  4. 2x2 − 7 = 0

  5. 3.27 + 18.5x = 2.29x2

A quadratic function is one whose highest-degree term is of second degree.

Example 2:

The following functions are quadratic functions:

  1. f(x) = 5x2 − 3x + 2

  2. f(x) = 9 − 3x2

  3. f(x) = x(x + 7)

  4. f(x) = x − 4 − 3x2

Some quadratic equations have a term missing. A quadratic that has no x term is called a pure quadratic; one that has no constant term is called an incomplete quadratic.

Example 3:

  1. x2 − 9 = 0 is a pure quadratic.

  2. x2 − 4x = 0 is an incomplete quadratic.

12.1.1. Solving a Quadratic Graphically

We will show several ways to solve a quadratic; first graphically and by calculator, and, in the next section, by formula.

▪ Exploration:

In our chapter on graphing we plotted the quadratic function

f(x) = x2 − 4x − 3

getting a curve that we called a parabola.

Try this. Either graph this function again or look back at our earlier graph. Does the curve intercept the x axis, and if so, how many times? Can you imagine a parabola that has more x-intercepts? Zoom out far enough to convince yourself that the curve will not ...

Get Technical Mathematics, Sixth Edition now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.