SOLUTION: please help factor and solve. 12a^2=5a+28 thank you

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: please help factor and solve. 12a^2=5a+28 thank you      Log On


   

Question 187006: please help factor and solve.

12a^2=5a+28

thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
12a%5E2=5a%2B28 Start with the given equation.


12a%5E2-5a-28=0 Get every term to the left side.

Now let's factor 12a%5E2-5a-28




Looking at the expression 12a%5E2-5a-28, we can see that the first coefficient is 12, the second coefficient is -5, and the last term is -28.


Now multiply the first coefficient 12 by the last term -28 to get %2812%29%28-28%29=-336.


Now the question is: what two whole numbers multiply to -336 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of -336 (the previous product).


Factors of -336:
1,2,3,4,6,7,8,12,14,16,21,24,28,42,48,56,84,112,168,336
-1,-2,-3,-4,-6,-7,-8,-12,-14,-16,-21,-24,-28,-42,-48,-56,-84,-112,-168,-336


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


These factors pair up and multiply to -336.
1*(-336)
2*(-168)
3*(-112)
4*(-84)
6*(-56)
7*(-48)
8*(-42)
12*(-28)
14*(-24)
16*(-21)
(-1)*(336)
(-2)*(168)
(-3)*(112)
(-4)*(84)
(-6)*(56)
(-7)*(48)
(-8)*(42)
(-12)*(28)
(-14)*(24)
(-16)*(21)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
1-3361+(-336)=-335
2-1682+(-168)=-166
3-1123+(-112)=-109
4-844+(-84)=-80
6-566+(-56)=-50
7-487+(-48)=-41
8-428+(-42)=-34
12-2812+(-28)=-16
14-2414+(-24)=-10
16-2116+(-21)=-5
-1336-1+336=335
-2168-2+168=166
-3112-3+112=109
-484-4+84=80
-656-6+56=50
-748-7+48=41
-842-8+42=34
-1228-12+28=16
-1424-14+24=10
-1621-16+21=5



From the table, we can see that the two numbers 16 and -21 add to -5 (the middle coefficient).


So the two numbers 16 and -21 both multiply to -336 and add to -5


Now replace the middle term -5a with 16a-21a. Remember, 16 and -21 add to -5. So this shows us that 16a-21a=-5a.


12a%5E2%2Bhighlight%2816a-21a%29-28 Replace the second term -5a with 16a-21a.


%2812a%5E2%2B16a%29%2B%28-21a-28%29 Group the terms into two pairs.


4a%283a%2B4%29%2B%28-21a-28%29 Factor out the GCF 4a from the first group.


4a%283a%2B4%29-7%283a%2B4%29 Factor out 7 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.


%284a-7%29%283a%2B4%29 Combine like terms. Or factor out the common term 3a%2B4


So 12a%5E2-5a-28 factors to %284a-7%29%283a%2B4%29.

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


So 12a%5E2-5a-28=0 then becomes %284a-7%29%283a%2B4%29=0 (after factoring the left side)


%284a-7%29%283a%2B4%29=0 Start with the given equation.


Now set each factor equal to zero (use the zero product property):


4a-7=0 or 3a%2B4=0


Now let's solve each individual equation:


4a-7=0 Start with the first equation.


4a=0%2B7 Add 7 to both sides.


4a=7 Combine like terms on the right side.


a=7%2F4 Divide both sides by 4 to isolate a. This is one of the solutions.

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


3a%2B4=0 Move onto the second equation.


3a=0-4 Subtract 4 from both sides.


3a=-4 Combine like terms on the right side.


a=-4%2F3 Divide both sides by 3 to isolate a. This is the other solution.


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

Answer:


So the solutions are a=7%2F4 or a=-4%2F3