SOLUTION: please factor and solve 12z^2+z=6

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: please factor and solve 12z^2+z=6      Log On


   

Question 187009: please factor and solve

12z^2+z=6
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
12z%5E2%2Bz=6 Start with the given equation.


12z%5E2%2Bz-6=0 Subtract 6 from both sides.


Let's factor the left side:


Looking at the expression 12z%5E2%2Bz-6, we can see that the first coefficient is 12, the second coefficient is 1, and the last term is -6.


Now multiply the first coefficient 12 by the last term -6 to get %2812%29%28-6%29=-72.


Now the question is: what two whole numbers multiply to -72 (the previous product) and add to the second coefficient 1?


To find these two numbers, we need to list all of the factors of -72 (the previous product).


Factors of -72:
1,2,3,4,6,8,9,12,18,24,36,72
-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72


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


These factors pair up and multiply to -72.
1*(-72)
2*(-36)
3*(-24)
4*(-18)
6*(-12)
8*(-9)
(-1)*(72)
(-2)*(36)
(-3)*(24)
(-4)*(18)
(-6)*(12)
(-8)*(9)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:


First NumberSecond NumberSum
1-721+(-72)=-71
2-362+(-36)=-34
3-243+(-24)=-21
4-184+(-18)=-14
6-126+(-12)=-6
8-98+(-9)=-1
-172-1+72=71
-236-2+36=34
-324-3+24=21
-418-4+18=14
-612-6+12=6
-89-8+9=1



From the table, we can see that the two numbers -8 and 9 add to 1 (the middle coefficient).


So the two numbers -8 and 9 both multiply to -72 and add to 1


Now replace the middle term 1z with -8z%2B9z. Remember, -8 and 9 add to 1. So this shows us that -8z%2B9z=1z.


12z%5E2%2Bhighlight%28-8z%2B9z%29-6 Replace the second term 1z with -8z%2B9z.


%2812z%5E2-8z%29%2B%289z-6%29 Group the terms into two pairs.


4z%283z-2%29%2B%289z-6%29 Factor out the GCF 4z from the first group.


4z%283z-2%29%2B3%283z-2%29 Factor out 3 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.


%284z%2B3%29%283z-2%29 Combine like terms. Or factor out the common term 3z-2


So 12z%5E2%2Bz-6 factors to %284z%2B3%29%283z-2%29.


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


So 12z%5E2%2Bz-6=0 becomes %284z%2B3%29%283z-2%29=0


Now set each factor equal to zero:
4z%2B3=0 or 3z-2=0

z=-3%2F4 or z=2%2F3 Now solve for z in each case


So our answers are

z=-3%2F4 or z=2%2F3