SOLUTION: solve algebraically using only one variable: the length of a rectangle is two more than twice its width. if the area of the rectangle is 84, find the length and the width.

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Question 228533: solve algebraically using only one variable: the length of a rectangle is two more than twice its width. if the area of the rectangle is 84, find the length and the width.
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Solve algebraically using only one variable: the length of a rectangle is two more than twice its width. If the area of the rectangle is 84, find the length and the width.
Step 1. Area A of a rectangle is A=width times the length

Step 2. Let w be the width.

Step 3. Let 2w+2 be the length since the length it's two more than twice its width.

Step 4. Then, A=w%2A%282w%2B2%29=2w%5E2%2B2w=84.

Step 5. Solving yields the following steps:

Subtract 84 from both sides of the equation

2w%5E2%2B2w-84=84-84

2w%5E2%2B2w-84=0

Step 6. To solve, use the quadratic equation given as

w+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=2, b=2, and c=-84.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case 2w%5E2%2B2w%2B-84+=+0) has the following solutons:

w%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A2%2A-84=676.

Discriminant d=676 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+676+%29%29%2F2%5Ca.

w%5B1%5D+=+%28-%282%29%2Bsqrt%28+676+%29%29%2F2%5C2+=+6
w%5B2%5D+=+%28-%282%29-sqrt%28+676+%29%29%2F2%5C2+=+-7

Quadratic expression 2w%5E2%2B2w%2B-84 can be factored:
2w%5E2%2B2w%2B-84+=+2%28w-6%29%2A%28w--7%29
Again, the answer is: 6, -7. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B2%2Ax%2B-84+%29



Selecting the positive solution

w=6 and 2w%2B2=14 and the A=6*14=84 which is a true statement.

Step 6. ANSWER: The width is 6 and the length is 14.

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

Good luck in your studies!

Respectfully,
Dr J