document.write( "Question 902240: Okay, I've asked this before and still don't understand.\r
\n" ); document.write( "\n" ); document.write( "Take the equationk\r
\n" ); document.write( "\n" ); document.write( " (3a² - 13a - 10)\r
\n" ); document.write( "\n" ); document.write( "=(3a ) (a )\r
\n" ); document.write( "\n" ); document.write( "How do I know which bracket gets the + and which bracket gets the -\r
\n" ); document.write( "\n" ); document.write( "How do I know whether the factors are 10 and 1 OR 5 and 2, seeing as both can give -10\r
\n" ); document.write( "\n" ); document.write( "So the book says the answer is (3a + 2 ) (a - 5)\r
\n" ); document.write( "\n" ); document.write( "So how do I know which bracket should get the 2 and which bracket gets the 5. I tried swapping them around and it didn't come to the quadratic equation.\r
\n" ); document.write( "\n" ); document.write( "Confused. What is the method of finding the solutions to these if the product of the outer terms IS NOT EQUAL to the middle term?\r
\n" ); document.write( "\n" ); document.write( "Confused\r
\n" ); document.write( "\n" ); document.write( "Kind Regards
\n" ); document.write( "Stefan\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #547170 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
First, it is not an equation. Equations equate with \"=\" It is an expression.
\n" ); document.write( "You have many good questions.
\n" ); document.write( "Let me know if this explanation helps or if you are still confused.
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"3a%5E2-13a-10\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"-10\"$ .

Now multiply the first coefficient $\"3\"$ by the last term $\"-10\"$ to get $\"%283%29%28-10%29=-30\"$ .

Now the question is: what two whole numbers multiply to $\"-30\"$ (the previous product) and add to the second coefficient $\"-13\"$ ?

To find these two numbers, we need to list all of the factors of $\"-30\"$ (the previous product).

Factors of $\"-30\"$ :

1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30

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

These factors pair up and multiply to $\"-30\"$ .

1*(-30) = -30
2*(-15) = -30
3*(-10) = -30
5*(-6) = -30
(-1)*(30) = -30
(-2)*(15) = -30
(-3)*(10) = -30
(-5)*(6) = -30

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-13\"$ :

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First Number	Second Number	Sum
1	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

From the table, we can see that the two numbers $\"2\"$ and $\"-15\"$ add to $\"-13\"$ (the middle coefficient).

So the two numbers $\"2\"$ and $\"-15\"$ both multiply to $\"-30\"$ and add to $\"-13\"$

Now replace the middle term $\"-13a\"$ with $\"2a-15a\"$ . Remember, $\"2\"$ and $\"-15\"$ add to $\"-13\"$ . So this shows us that $\"2a-15a=-13a\"$ .

$\"3a%5E2%2Bhighlight%282a-15a%29-10\"$ Replace the second term $\"-13a\"$ with $\"2a-15a\"$ .

$\"%283a%5E2%2B2a%29%2B%28-15a-10%29\"$ Group the terms into two pairs.

$\"a%283a%2B2%29%2B%28-15a-10%29\"$ Factor out the GCF $\"a\"$ from the first group.

$\"a%283a%2B2%29-5%283a%2B2%29\"$ Factor out $\"5\"$ 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.

$\"%28a-5%29%283a%2B2%29\"$ Combine like terms. Or factor out the common term $\"3a%2B2\"$

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Answer:

So $\"3%2Aa%5E2-13%2Aa-10\"$ factors to $\"%28a-5%29%283a%2B2%29\"$ .

In other words, $\"3%2Aa%5E2-13%2Aa-10=%28a-5%29%283a%2B2%29\"$ .

Note: you can check the answer by expanding $\"%28a-5%29%283a%2B2%29\"$ to get $\"3%2Aa%5E2-13%2Aa-10\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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