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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).