Solving Quadratic Equations

Let's examine the example from the previous page:

In that example, we had an equation of the form:

t² - 10t - 600 = 0

which 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.

What is actually meant by solving a quadratic equation?

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

Do you know how the graph for a quadratic equation look?
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.

Try to find the solutions of x² + 1 , x² + 2x + 1 , x² - 1 by using the Graphing workbench .