Lesson Three methods to find the dimensions of a rectangle when its area and the difference of two dimensions are given

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<H2>Three methods to find the dimensions of a rectangle when its area and the difference of two dimensions are given</H2>

<H3>Problem 1</H3>The area of a playground is &nbsp;168 &nbsp;square yards.
The width of the playground is &nbsp;2&nbsp; yards longer than its length. &nbsp;Find the length and width of the playground.

<U>1-st method</U>

First method is to create a corresponding quadratic equation and then solve it using the quadratic formula.

<pre>
Let  x  be the length of the rectangle, in yards. 
Then its width is (x-2) yards, according to the condition.

The area of the rectangle is

x*y = x*(x-2) = x^2 - 2x.

It is equal to 168 square yards, which gives you an equation

x^2 - 2x = 168,   or

x^2 - 2x - 168 = 0.      (1)

Apply the quadratic formula

{{{x[1,2]}}} = {{{(2 +- sqrt(2^2+4*168))/2}}} = {{{(2 +- sqrt(676))/2}}} = {{{(2 +- 26)/2}}} = {{{1 +- 13}}}.

You have two roots:  {{{x[1]}}} = 1 + 13 = 14  and  {{{x[2]}}} = 1 - 13 = -12.

Only positive root makes sense as the solution to this problem.

So, the length is 14 yards.

Then the width is  14-2 = 12 yards.

<U>Answer</U>.  The dimensions of the rectangle are  14 and  12  yards.
</pre>

Regarding this method, &nbsp;I'd like specially to highlight that the quadratic formula works for any quadratic equation.

It works like an army tank and provides a solution to any quadratic equation.

<U>2-nd method</U>

Second method is to create a corresponding quadratic equation and then solve it using factoring.

<pre>
Surely, the equation is the same as (1)

x^2 - 2x - 168 = 0,      (1)

and the method of obtaining this equation remains the same as in the <U>Solution 1</U>.

Now we want to factor the left side in the form

(x - ?)*(x - ?) = 0.       (2)

and now your goal is to find the numbers to put them instead of the question marks in order for to get the same equation (1).

It not very difficult task. Your first wish is to try integer divisors of the number -168 whose sum is 2.

After some trials you will find them as -12 and 14.

Notice that their product is  (-12)*14 = -168  and their sum is  (-12)+14 = 2,

so (2) becomes equivalent to equation (1).

Now you get the same answer  x= 14 as in the <U>Solution 1</U>.
</pre>

Regarding this method, &nbsp;I'd like specially to highlight that it works good and fast when the coefficients of your original equation 
are small and good enough to that extent that you can mentally find the required combination quickly.

Another situation when it works good is when you know the solution in advance.

<U>3-rd method</U>

It is most beautiful method. &nbsp;It uses a &nbsp;<U>reduced</U> &nbsp;quadratic equation.

<B>Solution</B>

<pre>
Let x be the number (now unknown) which is exactly half way between the width and the length of the playground.

Then, as you understand,  the length = (x+1)  and  the width = (x-1).

The area = 168 = length*width = (x+1)*(x-1).

Thus (x+1)*(x-1) = 168,  which implies  {{{x^2 - 1}}} = 168,  i.e.  {{{x^2}}} = 168+1 = 169.

Hence, x = {{{sqrt(169)}}} = 13.

Then the length = 13+1 = 14  and the width = 13-1 = 12.
</pre>

You may find the third method in some recreational books.

Sometimes in math circles the teachers explain it to students.

In the school Math, &nbsp;the standard method of solution is to reduce the problem to a quadratic equation

and then solve it using the quadratic formula or factoring.

I presented the third method here to make your horizon wider.

If you demonstrate this method in the class, &nbsp;your teacher and your classmates will be impressed !

So,  &nbsp;it was as if you visited a Math circle today . . .

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