Gregory G.

asked • 09/05/16

College algebra math questions

Zachary wants to buy a rug for a room that is 12 ft wide and 15 ft long. He wants to leave a uniform strip of floor around the rug. He can afford to buy 108 ft squared of carpeting. What dimensions should the rug be? 

Gregory G.

Confused on the factoring portion. 
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09/05/16

Teresa G.

Hi Gregory,
 
This trinomial is the one that needs extra work , you can use trial and eroor or you can use factoring by regrouping.
 
If you use trial and error:  2x2 - 27x + 36  = 0  
 
Set-up two binomials   (                ) (                )  
the 2 first terms are the factors of 2x2  and that is just 2x and x   ( 2x       ) ( x        )
 
Now the operations between the two terms , Since the last term is positive , that means the 2 last terms will have the same sign, either both positive or both negative. How do you know what sign then,? check the middle term, since that is positive, so the sign must be both  negative ( - * - = +) 
 
(2x  -    )   ( x -      )   now you just have to find the last two terms, this where the trial and error will get in.  Think of two factors of 36 and plug it in those 2 last terms and FOIL it and see which one will give you 27  . The two factors of 36 that will work for your problem are 3 and 12. Note that order matters , so if one order didn't work, you will have to switch the numbers around to see if that works.  
 
so the two binomials would be  ( 2x - 3) ( x - 12)
 
You might have to do extra practice on factoring to master it. Factoring is a very powerful tool to solve a lot of problems so it is your worth your time to really master it.
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09/05/16

Teresa G.

TYPO HERE 
 
Now the operations between the two terms , Since the last term is positive , that means the 2 last terms will have the same sign, either both positive or both negative. How do you know what sign then,? check the middle term, since that is NEGATIVE , so the sign of the last two terms must be both negative ( - plus  - = -  which is the sign of the middle term) and (- * - = + which is the sign of the last term)  Looks complicated but it's not bad, just keep on practicing.
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09/05/16

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Teresa G. answered • 09/05/16

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Hi Gregory,
 
You always start out a problem by drawing a figure with all the information you have.
 
Imagine a rectangle 12ft  x 15ft. 
 
Let x = width of uniform strip around the rug
 
The dimensions of the rug , considering the uniform strip around would be:
 
L =  15 - 2x    
W =  12 - 2x
 
Solve for the width using the formula of Area of rectangle
 
L  x  W  = 108
(15 - 2x) (12 - 2x)  =   108                     Plug-in the values you have
180 - 30x - 24x  + 4x2  =  108               Multiply out the two binomials on the left side FOIL
180 - 54x + 4x2   =  108                        Combine like terms
4x2 - 54 x + 180 - 108  =  0                   Collect all terms on one side and set it equal to zero
4x2 - 54 x + 72      =   0                         Combine like terms ( We now have a quadratic equation and we will first try                                                                  to solve it by factoring)
2x2 - 27x  +  36     =  0                         Divide all terms by GCF's of all the terms ( 2)  so it will be easier to factor
(2x - 3) ( x - 12)     =  0                         Factor out. You have a trinomial so this will factor into 2 binomials. Set-up                                                                     your two sets of parenthesis, each parenthesis will have two terms. If you                                                                      need help with factoring this let me know.
 
2x - 3 = 0     or  x - 12 = 0                     Set each factor equal to zero using the Zero product Property
 
x = 3/2     or x = 12                                Solve for x. Now check for extraneous solution, a solution that doest make                                                                    sense to your problem
 
Upon checking you will see that 12 doesn't make sense because the width of your room is 12 so it is not possible to have a uniform strip that wide.
 
x = 3/2 = 1.5 ft = the widt of the uniform strip
 
Now we can solve for the dimensions of the rug.
 
L = 15 - 2x =  15 - 2(1.5)     =   12 ft
W = 12 - 2x  =  12 - 2(1.5)   =   9 ft.
 
The dimensions of the rug will be 9ft x 12ft and that has an area of 108 sq. ft.
 
 
 
 
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Karin S. answered • 09/05/16

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If you draw the room and make X the width of the strip of floor around the carpet, the area of the carpet becomes 
(15-2x)(12-2x) which you then set equal to 108. 
Simplifying we get:
180-54x+4x2=108. This can be simplified further by dividing by 2 and combining like terms to get 
 
2x2-27x+36=0. 
Using the ac method to factoring we have the factors of (36)(2)=72 that add to -27 as -24 and -3. So substituting them in we get
 
2x2-24x-3x+36=0. Factoring by grouping gives us
2x(x-12)-3(x-12)=0
so our factors become
(x-12)(2x-3)=0
 
solving we get x=12 or 3/2. we will rule out 12 bc our room is 12ft wide so that doesn't make any sense. So the length of the rug is 15-2(3/2)=12ft 
 
and the the width of the carpet is 12-2(3/2)=9 ft. 
Verifying the area of the carpet we check is (12)(9)=108.  
 
 
 
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