The rectangle given has a perimeter of 14 units. Find the value(s) of x by factoring a quadriatic equation.

one side= 12/(x+1)
other side= 6/(4-x)

Remember that the perimeter of a rectangle is: P = 2L + 2W

Note: P = perimeter, L = length, and W = width.

Therefore:

14 = 2[12/(x+1)] + 2[6/(4-x)]

14 = 24/(x+1) + 12/(4-x)

Common denominator is (x+1)(4-x).

14 = 24(4-x)/(x+1)(4-x) + 12(x+1)/(x+1)(4-x)

14 = {[24(4-x)] + [12(x+1)]}/(x+1)(4-x)

14 = (96 - 24x + 12x + 12)/(x+1)(4-x)

14 = (108 - 12x)/(x+1)(4-x)

Multiplying both sides by (x+1)(4-x) we have this result:

14(x+1)(4-x) = 108 - 12x

-14(x+1)(x-4) = 108 - 12x

-14(x^2 - 4x + x - 4) = 108 - 12x

-14(x^2 - 3x - 4) = 108 - 12x

-14x^2 + 42x + 56 = 108 - 12x

Set the equation equal to 0:

-14x^2 + 42x + 56 - 108 + 12x = 0

-14x^2 + 54x - 52 = 0

Factor out -2:

-2(7x^2 - 27x + 26) = 0

Then factor 7x^2 - 27x + 26:

-2(7x - 13)(x - 2) = 0

Set each factor in the parentheses equal to 0:

7x - 13 = 0; x = 13/7
x - 2 = 0; x = 2

Check these values with the original equation. It always helps to check your work!

I hope this will help with other problems of this type.

thank you soo much for all your help. I was just wondering if you can give me an outline of what steps to do for next time. such as:
find CD.
times both sides by CD.

thanks again.
Shay.

Sure, here is a step-by-step outline for solving a quadratic equation by factoring:

1. Write down the equation in standard form, with one side equal to zero.
2. Simplify the equation if needed.
3. Factor out any common factors from the equation.
4. Set each factor equal to zero and solve for x.
5. Check each solution by substituting it back into the original equation.
6. If the checked solutions are correct, then those are the final values of x.

I hope this helps, and if you have any further questions, feel free to ask!