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**Solving Quadratic Equations**

*Let's examine the example from the previous page:*In that example, we had an equation of the form:

**t**

^{2}- 10t - 600 = 0which we factored into two terms as follows:

**(t + 20) (t - 30)**

The answer would be *t = -20* and *t = 30*. Since -20 has no meaning,
the answer is simply *30*.

What if we cannot factor the equation?

Suppose that another project has a cost function of the form:

This equation cannot be factored as in the first example. One way to solve
this particular equation is by *completing the square*. We first move the 7
to the right hand side of the equal sign.

Then we proceed to add a number squared to both sides of the equation to complete
the square as follow:

At this point, the number has to be guessed.

When we reduce the equation, we get

The equation yields two answers

**If we cannot factor the equation, we can still solve it by the method of
completing the square as shown in the above example.**

**A few simple facts that you should know**

Did you know that there are other methods for solving a quadratic equation, such as

*factoring, completing the square*, or using the

*quadratic formula*?

How do I know which method to use?

- Use factoring when the equation is simple and the factors are obvious. Use
completing the square when you cannot factor the equation.
*When in doubt*, use the**Quadratic Formula**, shown on the next page, which works for any quadratic equations.

- Solving a quadratic equation means finding the values of x where the graph cuts the x-axis.

The graph for a quadratic equation is a

**parabola**.

- If the parabola cuts the x-axis at only one point it means that the quadratic equation has two solutions with the same value (the value of x where the parabola touches the x-axis).

- If the parabola cuts the x-axis at two points it means that it has two solutions (the points where the parabola crosses the x-axis).

- If the parabola doesn't cut the x-axis it means that the quadratic equation doesn't have any real solution.

^{2}+ 1 , x

^{2}+ 2x + 1 , x

^{2}- 1 by using the Graphing workbench .