SOLUTION: Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city's parks. The park is a rectangle w

Question 1133450: Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city's parks. The park is a rectangle with an area of
91x^2 + 57x − 18 m^2 as shown in the figure below. The length and width of the park are perfect factors of the area.
To easily determine the length and the width of the park, factor the area of the park by grouping.
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city's parks.
The park is a rectangle with an area of
91x^2 + 57x − 18 m^2 as shown in the figure below.
The length and width of the park are perfect factors of the area.
:
91 only has one pair of factors 13 & 7
(13x - __)(7x + __)
Factors of 18, 9 & 2 and 6 & 3, try them both,
(13x - 3)(7x + 6) seems to work, if you foil
91x^2 + 78x - 21x - 18 = 91x^2 + 57x - 18
then
(13x - 3) and (7x + 6) are the length and width of the park

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The response from the other tutor factors the expression for the area by trial and error, which is the appropriate method for this quadratic expression. This expression is not easily factored by grouping.

But let's try it, since that is what you were asked to do.

The most common (I think) method for factoring by grouping is to find the product of the leading coefficient and the constant term and then find two numbers with that same product whose difference is the coefficient of the linear term. (Note in doing this, my preference is to ignore the signs of the two factors, which means I am looking for the difference between the two factors.)

So in this method we also need to do some trial and error; but the kind of trial and error is more difficult than what is required in the other tutor's method for doing the factoring.

We have

91*18 = 2*3*3*7*13

as the prime factorization of the product of the leading coefficient and the constant term. Putting these factors together in different ways, we might have....

(2*13)(3*3*7) = 26*63 --> difference is 37, not what we want
(3*13)(2*3*7) = 39*42 --> difference is 3, not what we want

Note that in 91*18 the difference between the two factors is 73, which is more than we want; and in 63*26 the difference is 37, which is less than we want. That means we need a factorization with the smaller number greater than 18 and less than 26. The only such number we can make with the given prime factors is 21, so

(3*7)(2*3*13) = 21*78 --> difference is 57, which is what we want.

(You should recognize by now that this is a LOT more work than what is required by the other tutor's method...!)

Now, to complete the factoring by grouping, we go back to the original quadratic and rewrite it using two linear terms with the coefficients we have found.

RELATED QUESTIONS
Charlotte has appointed a chairperson to lead a city beautification project. The first... (answered by MathLover1,Alan3354)
Hi Charlotte and Lydia shared some cashew nuts in the ratio 6:7.When Charlotte gave 36... (answered by jorel555)
a company is creating two new divisions and four managers are eligable to be appointed... (answered by Boreal)
Hi Charlotte and Lydia shared cashew nuts in the ratio of 6:7.when Charlotte gave 36... (answered by stanbon,josmiceli,MathTherapy)
To paint the gym for homecoming, Senate needs 25 gallons of lead pait and has $141... (answered by stanbon)
The number of ways to choose 2 students from a class to be the chairperson and the vice... (answered by Boreal)
A chemist has two alloys, one of which is 15% gold and 15% lead and the other which is... (answered by josgarithmetic)
A chemist has two alloys, one of which is 15% gold and 10% lead and the other which is... (answered by ptaylor)
an artist makes three types of ceramic statues at a monthly cost of $650 for 180 statues. (answered by richwmiller)