College Algebra

Quadratic Applications

College Algebra: One Step at a Time

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Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL 32773

Extra Problem by Anthony in South Australia:

Given the area of a rectangle is 90m² and the perimeter of the rectangle is 40m, find the breadth of the rectangle.

Solution:

Let x = breadth (or width) of the rectangle. Since the perimeter of the rectangle is 40, you know that

Now, solve for L in terms of x:, by dividing both sides by 2:

Subtract L from each side:

Now, you can write the equation based upon the area of the rectangle:

Set the equation equal to zero, by adding and subtracting from each side:

Since this does NOT seem to factor, you must now solve this quadratic equation by either completing the square or quadratic formula

For completing the square method, see the end of this problem.

SOLUTION BY QUADRATIC FORMULA

To solve by the quadratic formula, remember that the solution to the quadratic equation:

is given by the formula:

In the equation:

Divide out the factor of 2:

The width (breadth) is the smaller of the dimensions



To find the length, remember







Notice that the length is the larger of the dimensions obtained when you solved for x.
FINAL ANSWER:    meters by meters

Check:                           Perimeter = 2W+2L

                                                           =

                                                            = + 20 + 2

                                                            = 40 m.                                                                                                                                                                                                            Area    =

                                                           =   )( )

                                                            =

                                                            = 90

SOLUTION BY COMPLETING THE SQUARE METHOD

When completing the square, you get the variables on the left side, and the number term on the right side of the equation. Also, the coefficient of the term must be 1. You need to add a number to each side of the equation in order to form a perfect square trinomial on the left side of the equation. This process is called “completing the square.”





Do you know about the “Half and Square” Rule? In order to complete the square, assuming that the coefficient is 1, you must take “half” of the coefficient of x, and “square”.   Half of is , and =100. You must add +100 to each side of the equation in order to express the left side as a perfect square trinomial.

Take the square root of each side. Don’t forget the “ ”!!

Add +10 to each side of the equation:



The width (breadth) is the smaller of the dimensions



To find the length, remember







Notice that the length is the larger of the dimensions obtained when you solved for x.

FINAL ANSWER:   meters by   meters

Check:                           Perimeter = 2W+2L

                                                           =

                                                            = + 20 + 2

                                                            = 40 m.

                                             Area    =

                                                           =   )( )

                                                            =

                                                            = 90

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Dr. Robert J. Rapalje	Altamonte Springs Campus
Contact me at: rapaljer@seminolestate.edu
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