Calculate the perimeter of the quadrilateral

Question
The perimeter of the quadrilateral is 120cm. Find the lengths and sides. (x+5)(x^2-3)(3x-8)(x+3).



Solution
Let us first define how to measure the perimeter of a quadrilateral. The perimeter of any quadrilateral, or any shape with four sides for that matter, is measured by adding the lengths of its sides. Hence, the quadrilateral on the right has a perimeter of 30 which is computed by adding the measure of its sides - 6 + 9 + 7 + 8.

Now, let us go back to our problem. The four sides of the quadrilateral are (x+5), (x^2-3), (3x-8) and (x+3). Hence, its perimeter is computed by (x+5) + (x^2-3) + (3x-8) + (x+3).

To solve for the length of the sides of the quadrilateral:

(x+5) + (x^2-3) + (3x-8) + (x+3) = 120
x^2 - 3+ x+5 + 3x - 8 + x + 3 = 120
x^2 + 5x - 3 = 120 ===> add like terms
x^2 + 5x - 123 = 0 ===> right side of the equation is set to zero
(x + ?) (x - ?) = 0 ====> to arrive at the root of the quadratic equation, think of two numbers that when you multiply them yields -123 and when you add them gives 5. Hmmm...difficult, however nothing prevents us to looking at a solution which is NOT a whole number.

Now, I got 13.87 and -8.87. Their multiple is -123.02 [close enough] and their sum is 5.

(x + 13.87) (x - 8.87) = 0
x = -13.87 ===> this is NOT an option because it is NEGATIVE.
x = 8.87

Now to calculate the lengths of the sides of the quadrilateral:
(x+5) ===> 13.87
(x^2-3) ==> 75.68
(3x-8) ===> 18.61
(x+3) ===> 11.87

Now to check whether the sum of the above is 120. 13.87 + 75.68 + 18.61 + 11.87 = 120.02 [close enough]

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