SOLUTION: Please solve by factoring or by using the quadratic formula: 3x^2-11x+6=0

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Question 380387: Please solve by factoring or by using the quadratic formula: 3x^2-11x+6=0
Answer by Jk22(389) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 3%2Ax%5E2-11%2Ax%2B6=0 ( notice a=3, b=-11, and c=6)





x+=+%28--11+%2B-+sqrt%28+%28-11%29%5E2-4%2A3%2A6+%29%29%2F%282%2A3%29 Plug in a=3, b=-11, and c=6




x+=+%2811+%2B-+sqrt%28+%28-11%29%5E2-4%2A3%2A6+%29%29%2F%282%2A3%29 Negate -11 to get 11




x+=+%2811+%2B-+sqrt%28+121-4%2A3%2A6+%29%29%2F%282%2A3%29 Square -11 to get 121 (note: remember when you square -11, you must square the negative as well. This is because %28-11%29%5E2=-11%2A-11=121.)




x+=+%2811+%2B-+sqrt%28+121%2B-72+%29%29%2F%282%2A3%29 Multiply -4%2A6%2A3 to get -72




x+=+%2811+%2B-+sqrt%28+49+%29%29%2F%282%2A3%29 Combine like terms in the radicand (everything under the square root)




x+=+%2811+%2B-+7%29%2F%282%2A3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%2811+%2B-+7%29%2F6 Multiply 2 and 3 to get 6


So now the expression breaks down into two parts


x+=+%2811+%2B+7%29%2F6 or x+=+%2811+-+7%29%2F6


Lets look at the first part:


x=%2811+%2B+7%29%2F6


x=18%2F6 Add the terms in the numerator

x=3 Divide


So one answer is

x=3




Now lets look at the second part:


x=%2811+-+7%29%2F6


x=4%2F6 Subtract the terms in the numerator

x=2%2F3 Divide


So another answer is

x=2%2F3


So our solutions are:

x=3 or x=2%2F3


Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 3x%5E2-11x%2B6, we can see that the first coefficient is 3, the second coefficient is -11, and the last term is 6.



Now multiply the first coefficient 3 by the last term 6 to get %283%29%286%29=18.



Now the question is: what two whole numbers multiply to 18 (the previous product) and add to the second coefficient -11?



To find these two numbers, we need to list all of the factors of 18 (the previous product).



Factors of 18:

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18



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



These factors pair up and multiply to 18.

1*18 = 18
2*9 = 18
3*6 = 18
(-1)*(-18) = 18
(-2)*(-9) = 18
(-3)*(-6) = 18


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -11:



First NumberSecond NumberSum
1181+18=19
292+9=11
363+6=9
-1-18-1+(-18)=-19
-2-9-2+(-9)=-11
-3-6-3+(-6)=-9




From the table, we can see that the two numbers -2 and -9 add to -11 (the middle coefficient).



So the two numbers -2 and -9 both multiply to 18 and add to -11



Now replace the middle term -11x with -2x-9x. Remember, -2 and -9 add to -11. So this shows us that -2x-9x=-11x.



3x%5E2%2Bhighlight%28-2x-9x%29%2B6 Replace the second term -11x with -2x-9x.



%283x%5E2-2x%29%2B%28-9x%2B6%29 Group the terms into two pairs.



x%283x-2%29%2B%28-9x%2B6%29 Factor out the GCF x from the first group.



x%283x-2%29-3%283x-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.



%28x-3%29%283x-2%29 Combine like terms. Or factor out the common term 3x-2



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



Answer:



So 3%2Ax%5E2-11%2Ax%2B6 factors to %28x-3%29%283x-2%29.



In other words, 3%2Ax%5E2-11%2Ax%2B6=%28x-3%29%283x-2%29.



Note: you can check the answer by expanding %28x-3%29%283x-2%29 to get 3%2Ax%5E2-11%2Ax%2B6 or by graphing the original expression and the answer (the two graphs should be identical).