Chapter 4.5, Problem 8LC

Solution

To Identify: The quadratic equation likely has exactly one solution by using tables.

A quadratic equation has exactly one solution if the range of the table contains only one zero.

There is no solution to the quadratic equation if the range of the table does not contain any zeros.

Given:

Using tables, how to recognize that a quadratic equation likely has exactly one solutions and no solutions.

Explanation:

By observing the table values, number of solutions can be found out for the corresponding quadratic equation.

A quadratic equation has exactly one solution if the range of the table contains only one zero.

Let's take an example:

$x$	$x^2-2x+1$
$-1$	$4$
$0$	$1$
$1$	$0$
$2$	$1$
$3$	$4$

Since there is only one zero in the range, which is at $x=1$ .The equation above has exactly one root.

Another example:

There is no solution to the quadratic equation if the range of the table does not contain any zeros.

Quadratic equation: $x^2+x+1=0$

$x$	$x^2+x+1=0$
$-1$	$1$
$0$	$1$
$1$	$3$
$2$	$7$
$3$	$13$

The range of the tabular values does not contain a zero when we expand the table for more values of $x$ . Thus, the quadratic equation has no zeros.