document.write( "Question 881459: \"The width of a rectangular room is 10 feet less than its length. The area of the room is 375 square feet. Find the width and length of the room.\"\r
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\n" ); document.write( "\n" ); document.write( "I started out with 2L+2(L-10)=357
\n" ); document.write( "2L+2L-20=357\r
\n" ); document.write( "\n" ); document.write( "4L=377\r
\n" ); document.write( "\n" ); document.write( "4L/4=377/4\r
\n" ); document.write( "\n" ); document.write( "L=94.25 square feet.\r
\n" ); document.write( "\n" ); document.write( "I don't feel like this is the right answer. I appreciate the help! Thanks!\r
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Algebra.Com's Answer #532227 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
Did you check it?
\n" ); document.write( "if 94.25 is the length what is the width?
\n" ); document.write( "84.25-10 is the width
\n" ); document.write( "94.25*84.25=7940.5625
\n" ); document.write( "you used the formula for circumference not area!\r
\n" ); document.write( "\n" ); document.write( "you want L*w=375
\n" ); document.write( "L*(L-10)=375
\n" ); document.write( "L^2-10L-375=0\r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2-10x-375\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-10\"$ , and the last term is $\"-375\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-375\"$ to get $\"%281%29%28-375%29=-375\"$ .

Now the question is: what two whole numbers multiply to $\"-375\"$ (the previous product) and add to the second coefficient $\"-10\"$ ?

To find these two numbers, we need to list all of the factors of $\"-375\"$ (the previous product).

Factors of $\"-375\"$ :

1,3,5,15,25,75,125,375

-1,-3,-5,-15,-25,-75,-125,-375

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"-375\"$ .

1*(-375) = -375
3*(-125) = -375
5*(-75) = -375
15*(-25) = -375
(-1)*(375) = -375
(-3)*(125) = -375
(-5)*(75) = -375
(-15)*(25) = -375

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-10\"$ :

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First Number	Second Number	Sum
1	-375	1+(-375)=-374
3	-125	3+(-125)=-122
5	-75	5+(-75)=-70
15	-25	15+(-25)=-10
-1	375	-1+375=374
-3	125	-3+125=122
-5	75	-5+75=70
-15	25	-15+25=10

From the table, we can see that the two numbers $\"15\"$ and $\"-25\"$ add to $\"-10\"$ (the middle coefficient).

So the two numbers $\"15\"$ and $\"-25\"$ both multiply to $\"-375\"$ and add to $\"-10\"$

Now replace the middle term $\"-10x\"$ with $\"15x-25x\"$ . Remember, $\"15\"$ and $\"-25\"$ add to $\"-10\"$ . So this shows us that $\"15x-25x=-10x\"$ .

$\"x%5E2%2Bhighlight%2815x-25x%29-375\"$ Replace the second term $\"-10x\"$ with $\"15x-25x\"$ .

$\"%28x%5E2%2B15x%29%2B%28-25x-375%29\"$ Group the terms into two pairs.

$\"x%28x%2B15%29%2B%28-25x-375%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x%2B15%29-25%28x%2B15%29\"$ Factor out $\"25\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

$\"%28x-25%29%28x%2B15%29\"$ Combine like terms. Or factor out the common term $\"x%2B15\"$

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Answer:

So $\"x%5E2-10%2Ax-375\"$ factors to $\"%28x-25%29%28x%2B15%29\"$ .

In other words, $\"x%5E2-10%2Ax-375=%28x-25%29%28x%2B15%29\"$ .

Note: you can check the answer by expanding $\"%28x-25%29%28x%2B15%29\"$ to get $\"x%5E2-10%2Ax-375\"$ or by graphing the original expression and the answer (the two graphs should be identical).

\n" ); document.write( "
\n" ); document.write( "the solver doesn't like L
\n" ); document.write( "discard negative answers
\n" ); document.write( "x-25=0
\n" ); document.write( "L=25
\n" ); document.write( "W=15
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