SOLUTION: Please find the quadratic formula of 6x^2+10x-4=0 thank you

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Question 256556: Please find the quadratic formula of 6x^2+10x-4=0

thank you
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Do you mean solve using the quadratic formula?
2(x+2)(3x-1) = 0
how to factor-
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


6%2Ax%5E2%2B10%2Ax-4 Start with the given expression.



2%283x%5E2%2B5x-2%29 Factor out the GCF 2.



Now let's try to factor the inner expression 3x%5E2%2B5x-2



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Looking at the expression 3x%5E2%2B5x-2, we can see that the first coefficient is 3, the second coefficient is 5, and the last term is -2.



Now multiply the first coefficient 3 by the last term -2 to get %283%29%28-2%29=-6.



Now the question is: what two whole numbers multiply to -6 (the previous product) and add to the second coefficient 5?



To find these two numbers, we need to list all of the factors of -6 (the previous product).



Factors of -6:

1,2,3,6

-1,-2,-3,-6



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



These factors pair up and multiply to -6.

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


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



First NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=1




From the table, we can see that the two numbers -1 and 6 add to 5 (the middle coefficient).



So the two numbers -1 and 6 both multiply to -6 and add to 5



Now replace the middle term 5x with -x%2B6x. Remember, -1 and 6 add to 5. So this shows us that -x%2B6x=5x.



3x%5E2%2Bhighlight%28-x%2B6x%29-2 Replace the second term 5x with -x%2B6x.



%283x%5E2-x%29%2B%286x-2%29 Group the terms into two pairs.



x%283x-1%29%2B%286x-2%29 Factor out the GCF x from the first group.



x%283x-1%29%2B2%283x-1%29 Factor out 2 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%2B2%29%283x-1%29 Combine like terms. Or factor out the common term 3x-1



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So 2%283x%5E2%2B5x-2%29 then factors further to 2%28x%2B2%29%283x-1%29



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Answer:



So 6%2Ax%5E2%2B10%2Ax-4 completely factors to 2%28x%2B2%29%283x-1%29.



In other words, 6%2Ax%5E2%2B10%2Ax-4=2%28x%2B2%29%283x-1%29.



Note: you can check the answer by expanding 2%28x%2B2%29%283x-1%29 to get 6%2Ax%5E2%2B10%2Ax-4 or by graphing the original expression and the answer (the two graphs should be identical).



the quadratic formula
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 6x%5E2%2B10x%2B-4+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2810%29%5E2-4%2A6%2A-4=196.

Discriminant d=196 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-10%2B-sqrt%28+196+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2810%29%2Bsqrt%28+196+%29%29%2F2%5C6+=+0.333333333333333
x%5B2%5D+=+%28-%2810%29-sqrt%28+196+%29%29%2F2%5C6+=+-2

Quadratic expression 6x%5E2%2B10x%2B-4 can be factored:
6x%5E2%2B10x%2B-4+=+6%28x-0.333333333333333%29%2A%28x--2%29
Again, the answer is: 0.333333333333333, -2. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B10%2Ax%2B-4+%29

2(x+2)(3x-1) = 0