SOLUTION: Please show me step by step how to arrive at the answer for the following expression: 6x^2-21x-45

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Please show me step by step how to arrive at the answer for the following expression: 6x^2-21x-45      Log On


   

Question 922569: Please show me step by step how to arrive at the answer for the following expression:
6x^2-21x-45
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor this.


6x%5E2-21x-45 Start with the given expression.


3%282x%5E2-7x-15%29 Factor out the GCF 3.


Now let's try to factor the inner expression 2x%5E2-7x-15


---------------------------------------------------------------


Looking at the expression 2x%5E2-7x-15, we can see that the first coefficient is 2, the second coefficient is -7, and the last term is -15.


Now multiply the first coefficient 2 by the last term -15 to get %282%29%28-15%29=-30.


Now the question is: what two whole numbers multiply to -30 (the previous product) and add to the second coefficient -7?


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 -7:


First NumberSecond NumberSum
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1



From the table, we can see that the two numbers 3 and -10 add to -7 (the middle coefficient).


So the two numbers 3 and -10 both multiply to -30 and add to -7


Now replace the middle term -7x with 3x-10x. Remember, 3 and -10 add to -7. So this shows us that 3x-10x=-7x.


2x%5E2%2Bhighlight%283x-10x%29-15 Replace the second term -7x with 3x-10x.


%282x%5E2%2B3x%29%2B%28-10x-15%29 Group the terms into two pairs.


x%282x%2B3%29%2B%28-10x-15%29 Factor out the GCF x from the first group.


x%282x%2B3%29-5%282x%2B3%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.


%28x-5%29%282x%2B3%29 Combine like terms. Or factor out the common term 2x%2B3


--------------------------------------------------


So 3%282x%5E2-7x-15%29 then factors further to 3%28x-5%29%282x%2B3%29


===============================================================


Answer:


So 6x%5E2-21x-45 completely factors to 3%28x-5%29%282x%2B3%29.


In other words, 6x%5E2-21x-45=3%28x-5%29%282x%2B3%29.


Note: you can check the answer by expanding 3%28x-5%29%282x%2B3%29 to get 6x%5E2-21x-45 or by graphing the original expression and the answer (the two graphs should be identical).


------------------------------------------------------------------------

Let me know if that helps or not. Thanks.

If you need more help, feel free to email me at jim_thompson5910@hotmail.com

My Website: http://www.freewebs.com/jimthompson5910/home.html