we are going to fence into a rectangular field & we know that for some cause we desire the field to have an enclosed area of 75 ft2. We also know that we desire the width of the field to be three feet longer than the length of the field. Determine dimensions of the field?

Solution

Thus, we'll assume x is the length of the field and so we know that x + 3 will be the width of the field.

Now, we also know that area of rectangle is length times width and thus we know that,

                                                   x ( x + 3) = 75

Now, it is a quadratic equation thus let's first writes it in standard form.

                                                   x² + 3x = 75

                                                x² + 3x - 75 = 0

Using the quadratic formula gives,

                                                x = (-3 ±√309)/2

Now, at this point, we've got to deal along with the fact that there are two solutions here and we only desire a single answer.  Thus, let's convert to decimals & see what the solutions really are.

x = (-3 + √309)/2 = 7.2892    and (-3 -√309)/2 = 7.2892 = -10.2892

Thus, we have one positive & one negative.  From the stand point of requiring the dimensions of field the negative solution doesn't make any sense thus we will avoid it.

Thus, the length of the field is 7.2892 feet. The width is three feet longer than this and thus is10.2892 feet.

Note that the width is almost the second solution to the quadratic equation. The only  single difference is the minus sign.  Do not expect this to always occur.  In this case it is more of a function of the problem.  For a more complexes set up this will NOT happen.

Now, from a physical point we can see that we have to expect to not get complex solutions to these problems.  Upon solving the quadratic equation we should get either two real distinct solutions or a double root. Also, as the previous example has illustrated, when we get two real distinct solutions we will be able to eliminate one of them for physical reasons.