document.write( "Question 909572: Question: A rectangle has an area of 330 meters squared. one side is 7 meters longer than the other. what are the dimensions of the rectangle?\r
\n" ); document.write( "\n" ); document.write( "My Attempt: x(x+7) = 330
\n" ); document.write( " x^2 + 7x = 330
\n" ); document.write( "square root both sides x + 7x = 18.2
\n" ); document.write( " 8x/8 = 18.2/8
\n" ); document.write( " x = 2.25
\n" ); document.write( "It's wrong. \n" ); document.write( "

Algebra.Com's Answer #551923 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
x + 7x = 18.2
\n" ); document.write( "How is x + 7x the same as $\"sqrt%28x%5E2+%2B+7x%29+\"$ ?
\n" ); document.write( "x^2 + 7x = 330
\n" ); document.write( "x^2 + 7x - 330=0
\n" ); document.write( "Try to factor
\n" ); document.write( "Dismiss negative answers for measurements.
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2%2B7x-330\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"7\"$ , and the last term is $\"-330\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-330\"$ to get $\"%281%29%28-330%29=-330\"$ .

Now the question is: what two whole numbers multiply to $\"-330\"$ (the previous product) and add to the second coefficient $\"7\"$ ?

To find these two numbers, we need to list all of the factors of $\"-330\"$ (the previous product).

Factors of $\"-330\"$ :

1,2,3,5,6,10,11,15,22,30,33,55,66,110,165,330

-1,-2,-3,-5,-6,-10,-11,-15,-22,-30,-33,-55,-66,-110,-165,-330

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

These factors pair up and multiply to $\"-330\"$ .

1*(-330) = -330
2*(-165) = -330
3*(-110) = -330
5*(-66) = -330
6*(-55) = -330
10*(-33) = -330
11*(-30) = -330
15*(-22) = -330
(-1)*(330) = -330
(-2)*(165) = -330
(-3)*(110) = -330
(-5)*(66) = -330
(-6)*(55) = -330
(-10)*(33) = -330
(-11)*(30) = -330
(-15)*(22) = -330

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"7\"$ :

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First Number	Second Number	Sum
1	-330	1+(-330)=-329
2	-165	2+(-165)=-163
3	-110	3+(-110)=-107
5	-66	5+(-66)=-61
6	-55	6+(-55)=-49
10	-33	10+(-33)=-23
11	-30	11+(-30)=-19
15	-22	15+(-22)=-7
-1	330	-1+330=329
-2	165	-2+165=163
-3	110	-3+110=107
-5	66	-5+66=61
-6	55	-6+55=49
-10	33	-10+33=23
-11	30	-11+30=19
-15	22	-15+22=7

From the table, we can see that the two numbers $\"-15\"$ and $\"22\"$ add to $\"7\"$ (the middle coefficient).

So the two numbers $\"-15\"$ and $\"22\"$ both multiply to $\"-330\"$ and add to $\"7\"$

Now replace the middle term $\"7x\"$ with $\"-15x%2B22x\"$ . Remember, $\"-15\"$ and $\"22\"$ add to $\"7\"$ . So this shows us that $\"-15x%2B22x=7x\"$ .

$\"x%5E2%2Bhighlight%28-15x%2B22x%29-330\"$ Replace the second term $\"7x\"$ with $\"-15x%2B22x\"$ .

$\"%28x%5E2-15x%29%2B%2822x-330%29\"$ Group the terms into two pairs.

$\"x%28x-15%29%2B%2822x-330%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x-15%29%2B22%28x-15%29\"$ Factor out $\"22\"$ 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.

$\"%28x%2B22%29%28x-15%29\"$ Combine like terms. Or factor out the common term $\"x-15\"$

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Answer:

So $\"x%5E2%2B7%2Ax-330\"$ factors to $\"%28x%2B22%29%28x-15%29\"$ .

In other words, $\"x%5E2%2B7%2Ax-330=%28x%2B22%29%28x-15%29\"$ .

Note: you can check the answer by expanding $\"%28x%2B22%29%28x-15%29\"$ to get $\"x%5E2%2B7%2Ax-330\"$ or by graphing the original expression and the answer (the two graphs should be identical).

\n" ); document.write( "\n" ); document.write( "If not use quadratic formula
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Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation $\"ax%5E2%2Bbx%2Bc=0\"$ (in our case $\"1x%5E2%2B7x%2B-330+=+0\"$ ) has the following solutons:
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\n" ); document.write( " $\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"$
\n" ); document.write( "
\n" ); document.write( " For these solutions to exist, the discriminant $\"b%5E2-4ac\"$ should not be a negative number.
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\n" ); document.write( " First, we need to compute the discriminant $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%287%29%5E2-4%2A1%2A-330=1369\"$ .
\n" ); document.write( "
\n" ); document.write( " Discriminant d=1369 is greater than zero. That means that there are two solutions: $\"+x%5B12%5D+=+%28-7%2B-sqrt%28+1369+%29%29%2F2%5Ca\"$ .
\n" ); document.write( "
\n" ); document.write( " $\"x%5B1%5D+=+%28-%287%29%2Bsqrt%28+1369+%29%29%2F2%5C1+=+15\"$
\n" ); document.write( " $\"x%5B2%5D+=+%28-%287%29-sqrt%28+1369+%29%29%2F2%5C1+=+-22\"$
\n" ); document.write( "
\n" ); document.write( " Quadratic expression $\"1x%5E2%2B7x%2B-330\"$ can be factored:
\n" ); document.write( " $\"1x%5E2%2B7x%2B-330+=+1%28x-15%29%2A%28x--22%29\"$
\n" ); document.write( " Again, the answer is: 15, -22.\n" ); document.write( "Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B7%2Ax%2B-330+%29\"$

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