SOLUTION: Regarding algebfra, polynomials -- Please help me solve this problem. I know the answer is x = 5, but I'm having trouble arriving at the answer correctly. "The area of a rec

Question 730838: Regarding algebfra, polynomials --
Please help me solve this problem. I know the answer is x = 5, but I'm having trouble arriving at the answer correctly.
"The area of a rectangle is 55 square feet. The height is x and the length is 2x + 1. Solve for x."

2x^2 + x = 55

2x^2 + x -55 = 0

My email address is mjj1107@sbcglobal.net.
Thanks.
Mary Jane
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
You are on the right path and just need to solve
There are easier quadratic equations

I know of 3 ways to solve quadratic equations like
Factoring will work if the answers are rational numbers.
Completing the square will always work.
Using the quadratic formula will always work too.

FACTORING:
When the leading coefficient is not 1 or -1, factoring is a little harder.
In this case, you must look for pairs of factors of .
Giving a negative sign to one factor and a positive sign to the other, they must add up to the coefficient of the term in x, 1.
The number 110 can be written as 4 different products:

The last one is the one that works because
So we use 11 and -10 as coefficients of x and write as

Then we factor by grouping, like this:

So, since we re-write the equation as
and find the solutions that make
--> and
-->
Since x must be positive to be the width of a rectangle, the only solution is .

COMPLETING THE SQUARE:
--> dividing both sides by 2
is part of so if we add to both sides we "complete the square:
--> --> -->
or so
either --> --> -->
or --> --> -->
Same solutions to the equation, and the only solution to the geometry problem is .

THE QUADRATIC FORMULA
is a formula that derives from completing the square.
I never set to memorize it, but I have been using it for so long that I remember it.
For an equation of the form
the solutions are given by the quadratic formula:

In the case of , and so

So the solutions of the equation are
--> --> --> -->
Since cannot be the with of a rectangle, the only solution is .
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