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 ...

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