Question 902240: Okay, I've asked this before and still don't understand.
Take the equationk
(3aČ - 13a - 10)
=(3a ) (a )
How do I know which bracket gets the + and which bracket gets the -
How do I know whether the factors are 10 and 1 OR 5 and 2, seeing as both can give -10
So the book says the answer is (3a + 2 ) (a - 5)
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.
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?
Confused
Kind Regards
Stefan
Found 2 solutions by richwmiller, stanbon:
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! First, it is not an equation. Equations equate with "=" It is an expression.
You have many good questions.
Let me know if this explanation helps or if you are still confused.
| Solved by pluggable solver: Factoring using the AC method (Factor by Grouping) |
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to  (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
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 .
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 :
| 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  and  add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out  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.
 Combine like terms. Or factor out the common term 
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Answer:
So  factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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Answer by stanbon(75887)  (Show Source):
You can put this solution on YOUR website! Okay, I've asked this before and still don't understand.
Take the equationk
(3aČ - 13a - 10)
=(3a ) (a )
How do I know which bracket gets the + and which bracket gets the -
How do I know whether the factors are 10 and 1 OR 5 and 2, seeing as both can give -10
So the book says the answer is (3a + 2 ) (a - 5)
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.
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Use the AC Method to factor the quadratic::
=====
(3aČ - 13a - 10)
AC = 3*-10 = -30
B = -13
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Find 2 number whose product is AC = -30 and whose sum is -13::
Ans:: 15*-2 = 30
----
Substitute for B
3a^2 -15a + 2a - 10
---
Factor to get:
3a(a-5)+2(a-5)
Factor again to get:
(a-5)(3a+2)
---------------
What is the method of finding the solutions to these if the product of the outer terms IS NOT EQUAL to the middle term?
Ans:: Use the quadratic formula to aolve ANY quadratic::
x = [-b +- sqrt(b^2-4ac)]/(2a)
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Cheers,
Stan H.
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