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Solving Quadratic Equations

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

Solve the quadratic equation

.

Solution


 

The Quadratic Formula

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).