document.write( "Question 862389: A person rowed her boat upstream for a distance of 32 miles and then rowed backed to the starting point. The total time of the trip was 18 hours. If the rate of the current was 7 mph, find the average of the boat in still water.\r
\n" ); document.write( "\n" ); document.write( "My daughter and I are stuck on this problem, many thanks. \n" ); document.write( "

Algebra.Com's Answer #519703 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
Both of the other tutors started out fine but made errors.\r
\n" ); document.write( "\n" ); document.write( "There is a simple answer.
\n" ); document.write( "What grade is your daughter in?
\n" ); document.write( "18 = (64 b)/((b-7) (b+7))\r
\n" ); document.write( "\n" ); document.write( "b=9
\n" ); document.write( "the boat goes 9 mph
\n" ); document.write( "One started out fine but made some simple errors
\n" ); document.write( "18 = (64 b)/(b^2-49)
\n" ); document.write( "18(b^2-49)-64b=0
\n" ); document.write( "18b^2-64b-49*18=0
\n" ); document.write( "18b^2-64b-882=0
\n" ); document.write( "using the quadratic equation
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Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation $\"ax%5E2%2Bbx%2Bc=0\"$ (in our case $\"18x%5E2%2B-64x%2B-882+=+0\"$ ) has the following solutons:
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\n" ); document.write( " $\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"$
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\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 $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%28-64%29%5E2-4%2A18%2A-882=67600\"$ .
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\n" ); document.write( " Discriminant d=67600 is greater than zero. That means that there are two solutions: $\"+x%5B12%5D+=+%28--64%2B-sqrt%28+67600+%29%29%2F2%5Ca\"$ .
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\n" ); document.write( " $\"x%5B1%5D+=+%28-%28-64%29%2Bsqrt%28+67600+%29%29%2F2%5C18+=+9\"$
\n" ); document.write( " $\"x%5B2%5D+=+%28-%28-64%29-sqrt%28+67600+%29%29%2F2%5C18+=+-5.44444444444444\"$
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\n" ); document.write( " Quadratic expression $\"18x%5E2%2B-64x%2B-882\"$ can be factored:
\n" ); document.write( " $\"18x%5E2%2B-64x%2B-882+=+18%28x-9%29%2A%28x--5.44444444444444%29\"$
\n" ); document.write( " Again, the answer is: 9, -5.44444444444444.\n" ); document.write( "Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+18%2Ax%5E2%2B-64%2Ax%2B-882+%29\"$

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\n" ); document.write( "\n" ); document.write( "and factoring
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"18%2Ax%5E2-64%2Ax-882\"$ Start with the given expression.

$\"2%289x%5E2-32x-441%29\"$ Factor out the GCF $\"2\"$ .

Now let's try to factor the inner expression $\"9x%5E2-32x-441\"$

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Looking at the expression $\"9x%5E2-32x-441\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"-32\"$ , and the last term is $\"-441\"$ .

Now multiply the first coefficient $\"9\"$ by the last term $\"-441\"$ to get $\"%289%29%28-441%29=-3969\"$ .

Now the question is: what two whole numbers multiply to $\"-3969\"$ (the previous product) and add to the second coefficient $\"-32\"$ ?

To find these two numbers, we need to list all of the factors of $\"-3969\"$ (the previous product).

Factors of $\"-3969\"$ :

1,3,7,9,21,27,49,63,81,147,189,441,567,1323,3969

-1,-3,-7,-9,-21,-27,-49,-63,-81,-147,-189,-441,-567,-1323,-3969

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

These factors pair up and multiply to $\"-3969\"$ .

1*(-3969) = -3969
3*(-1323) = -3969
7*(-567) = -3969
9*(-441) = -3969
21*(-189) = -3969
27*(-147) = -3969
49*(-81) = -3969
63*(-63) = -3969
(-1)*(3969) = -3969
(-3)*(1323) = -3969
(-7)*(567) = -3969
(-9)*(441) = -3969
(-21)*(189) = -3969
(-27)*(147) = -3969
(-49)*(81) = -3969
(-63)*(63) = -3969

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-32\"$ :

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First Number	Second Number	Sum
1	-3969	1+(-3969)=-3968
3	-1323	3+(-1323)=-1320
7	-567	7+(-567)=-560
9	-441	9+(-441)=-432
21	-189	21+(-189)=-168
27	-147	27+(-147)=-120
49	-81	49+(-81)=-32
63	-63	63+(-63)=0
-1	3969	-1+3969=3968
-3	1323	-3+1323=1320
-7	567	-7+567=560
-9	441	-9+441=432
-21	189	-21+189=168
-27	147	-27+147=120
-49	81	-49+81=32
-63	63	-63+63=0

From the table, we can see that the two numbers $\"49\"$ and $\"-81\"$ add to $\"-32\"$ (the middle coefficient).

So the two numbers $\"49\"$ and $\"-81\"$ both multiply to $\"-3969\"$ and add to $\"-32\"$

Now replace the middle term $\"-32x\"$ with $\"49x-81x\"$ . Remember, $\"49\"$ and $\"-81\"$ add to $\"-32\"$ . So this shows us that $\"49x-81x=-32x\"$ .

$\"9x%5E2%2Bhighlight%2849x-81x%29-441\"$ Replace the second term $\"-32x\"$ with $\"49x-81x\"$ .

$\"%289x%5E2%2B49x%29%2B%28-81x-441%29\"$ Group the terms into two pairs.

$\"x%289x%2B49%29%2B%28-81x-441%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%289x%2B49%29-9%289x%2B49%29\"$ Factor out $\"9\"$ 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.

$\"%28x-9%29%289x%2B49%29\"$ Combine like terms. Or factor out the common term $\"9x%2B49\"$

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So $\"2%289x%5E2-32x-441%29\"$ then factors further to $\"2%28x-9%29%289x%2B49%29\"$

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Answer:

So $\"18%2Ax%5E2-64%2Ax-882\"$ completely factors to $\"2%28x-9%29%289x%2B49%29\"$ .

In other words, $\"18%2Ax%5E2-64%2Ax-882=2%28x-9%29%289x%2B49%29\"$ .

Note: you can check the answer by expanding $\"2%28x-9%29%289x%2B49%29\"$ to get $\"18%2Ax%5E2-64%2Ax-882\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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