document.write( "Question 894032: In a two digit number, the ten's digit is bigger. The product of the digits is 27 and the difference between two digits is 6. Find the number...
\n" ); document.write( "Answer is given i.e. 93.
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Algebra.Com's Answer #541695 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
n=10t+o
\n" ); document.write( "t>o
\n" ); document.write( "t*o=27
\n" ); document.write( "t-o=6
\n" ); document.write( "t=o+6
\n" ); document.write( "t*o=27
\n" ); document.write( "o*(o+6)=27
\n" ); document.write( "o^2+6o-27=0
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"o%5E2%2B6o-27\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"6\"$ , and the last term is $\"-27\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-27\"$ to get $\"%281%29%28-27%29=-27\"$ .

Now the question is: what two whole numbers multiply to $\"-27\"$ (the previous product) and add to the second coefficient $\"6\"$ ?

To find these two numbers, we need to list all of the factors of $\"-27\"$ (the previous product).

Factors of $\"-27\"$ :

1,3,9,27

-1,-3,-9,-27

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

These factors pair up and multiply to $\"-27\"$ .

1*(-27) = -27
3*(-9) = -27
(-1)*(27) = -27
(-3)*(9) = -27

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"6\"$ :

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First Number	Second Number	Sum
1	-27	1+(-27)=-26
3	-9	3+(-9)=-6
-1	27	-1+27=26
-3	9	-3+9=6

From the table, we can see that the two numbers $\"-3\"$ and $\"9\"$ add to $\"6\"$ (the middle coefficient).

So the two numbers $\"-3\"$ and $\"9\"$ both multiply to $\"-27\"$ and add to $\"6\"$

Now replace the middle term $\"6o\"$ with $\"-3o%2B9o\"$ . Remember, $\"-3\"$ and $\"9\"$ add to $\"6\"$ . So this shows us that $\"-3o%2B9o=6o\"$ .

$\"o%5E2%2Bhighlight%28-3o%2B9o%29-27\"$ Replace the second term $\"6o\"$ with $\"-3o%2B9o\"$ .

$\"%28o%5E2-3o%29%2B%289o-27%29\"$ Group the terms into two pairs.

$\"o%28o-3%29%2B%289o-27%29\"$ Factor out the GCF $\"o\"$ from the first group.

$\"o%28o-3%29%2B9%28o-3%29\"$ 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.

$\"%28o%2B9%29%28o-3%29\"$ Combine like terms. Or factor out the common term $\"o-3\"$

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Answer:

So $\"o%5E2%2B6%2Ao-27\"$ factors to $\"%28o%2B9%29%28o-3%29\"$ .

In other words, $\"o%5E2%2B6%2Ao-27=%28o%2B9%29%28o-3%29\"$ .

Note: you can check the answer by expanding $\"%28o%2B9%29%28o-3%29\"$ to get $\"o%5E2%2B6%2Ao-27\"$ or by graphing the original expression and the answer (the two graphs should be identical).

\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "o=3
\n" ); document.write( "t-3=6
\n" ); document.write( "t=9
\n" ); document.write( "n=93 \n" ); document.write( "

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