                   ### Factoring Method

A quadratic equation has the form If the expression can be factored, then we can set each factor equal to zero. The following example illustrates the factoring method.

### Example .

### Solution Since most quadratic expressions cannot be factored, the factoring method is limited in its usefulness. However, any quadratic equation that has real solutions can be solved by the quadratic formula The quantity is called the discriminant of the quadratic equation because it determines whether the equation has one, two, or no real solutions:

If , the equation has two real solutions.

If , the equation has exactly one solution.

If , the equation has no real solutions.

### Example

Use the quadratic formula to find the solutions, if any, of the quadratic equation ### Solution

Comparing our equation to , we see that Substituting these values into the quadratic formula we get Evaluating this expression, we find two solutions, x = -5.49 and x =1.17 (rounded to two decimal places).

### Example

Use the quadratic formula to find the solutions, if any, of the equation ### Solution

In this case, we must first subtract 160 from both sides so that the equation is in the form . We now see that Substituting these values into the quadratic formula, we get, .

Evaluating this expression yields the two solutions t= -0.71 and t = 3.52 (rounded to two decimal places).  