SOLUTION: Solve the equation 3x2 - 15x = 0. 6x2 - 5x = 6 Factor the expression x2 - 8xy + 12y2 completely. Choose one factor of the following expression from the list b

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Question 147136: Solve the equation 3x2 - 15x = 0.

6x2 - 5x = 6


Factor the expression x2 - 8xy + 12y2 completely.


Choose one factor of the following expression from the list below.
6x2 - 5x - 4

Thanks for the help and i will also put a donation for you.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1: Solve the equation 3x2 - 15x = 0.



Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-15%29+%2B-+sqrt%28+%28-15%29%5E2-4%283%29%280%29+%29%29%2F%282%283%29%29 Plug in a=3, b=-15, and c=0


x+=+%2815+%2B-+sqrt%28+%28-15%29%5E2-4%283%29%280%29+%29%29%2F%282%283%29%29 Negate -15 to get 15.


x+=+%2815+%2B-+sqrt%28+225-4%283%29%280%29+%29%29%2F%282%283%29%29 Square -15 to get 225.


x+=+%2815+%2B-+sqrt%28+225-0+%29%29%2F%282%283%29%29 Multiply 4%283%29%280%29 to get 0


x+=+%2815+%2B-+sqrt%28+225+%29%29%2F%282%283%29%29 Subtract 0 from 225 to get 225


x+=+%2815+%2B-+sqrt%28+225+%29%29%2F%286%29 Multiply 2 and 3 to get 6.


x+=+%2815+%2B-+15%29%2F%286%29 Take the square root of 225 to get 15.


x+=+%2815+%2B+15%29%2F%286%29 or x+=+%2815+-+15%29%2F%286%29 Break up the expression.


x+=+%2830%29%2F%286%29 or x+=++%280%29%2F%286%29 Combine like terms.


x+=+5 or x+=+0 Simplify.


So our answers are x+=+5 or x+=+0





# 2: Solve the equation 6x2 - 5x = 6





6x%5E2-5x=6 Start with the given equation.


6x%5E2-5x-6=0 Get all terms to the left side.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=6, b=-5, and c=-6


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%286%29%28-6%29+%29%29%2F%282%286%29%29 Plug in a=6, b=-5, and c=-6


x+=+%285+%2B-+sqrt%28+%28-5%29%5E2-4%286%29%28-6%29+%29%29%2F%282%286%29%29 Negate -5 to get 5.


x+=+%285+%2B-+sqrt%28+25-4%286%29%28-6%29+%29%29%2F%282%286%29%29 Square -5 to get 25.


x+=+%285+%2B-+sqrt%28+25--144+%29%29%2F%282%286%29%29 Multiply 4%286%29%28-6%29 to get -144


x+=+%285+%2B-+sqrt%28+25%2B144+%29%29%2F%282%286%29%29 Rewrite sqrt%2825--144%29 as sqrt%2825%2B144%29


x+=+%285+%2B-+sqrt%28+169+%29%29%2F%282%286%29%29 Add 25 to 144 to get 169


x+=+%285+%2B-+sqrt%28+169+%29%29%2F%2812%29 Multiply 2 and 6 to get 12.


x+=+%285+%2B-+13%29%2F%2812%29 Take the square root of 169 to get 13.


x+=+%285+%2B+13%29%2F%2812%29 or x+=+%285+-+13%29%2F%2812%29 Break up the expression.


x+=+%2818%29%2F%2812%29 or x+=++%28-8%29%2F%2812%29 Combine like terms.


x+=+3%2F2 or x+=+-2%2F3 Simplify.


So our answers are x+=+3%2F2 or x+=+-2%2F3






# 3: Factor the expression x2 - 8xy + 12y2 completely.




Looking at 1x%5E2-8xy%2B12y%5E2 we can see that the first term is 1x%5E2 and the last term is 12y%5E2 where the coefficients are 1 and 12 respectively.

Now multiply the first coefficient 1 and the last coefficient 12 to get 12. Now what two numbers multiply to 12 and add to the middle coefficient -8? Let's list all of the factors of 12:



Factors of 12:
1,2,3,4,6,12

-1,-2,-3,-4,-6,-12 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 12
1*12
2*6
3*4
(-1)*(-12)
(-2)*(-6)
(-3)*(-4)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -8? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -8

First NumberSecond NumberSum
1121+12=13
262+6=8
343+4=7
-1-12-1+(-12)=-13
-2-6-2+(-6)=-8
-3-4-3+(-4)=-7



From this list we can see that -2 and -6 add up to -8 and multiply to 12


Now looking at the expression 1x%5E2-8xy%2B12y%5E2, replace -8xy with -2xy%2B-6xy (notice -2xy%2B-6xy adds up to -8xy. So it is equivalent to -8xy)

1x%5E2%2Bhighlight%28-2xy%2B-6xy%29%2B12y%5E2


Now let's factor 1x%5E2-2xy-6xy%2B12y%5E2 by grouping:


%281x%5E2-2xy%29%2B%28-6xy%2B12y%5E2%29 Group like terms


x%28x-2y%29-6y%28x-2y%29 Factor out the GCF of x out of the first group. Factor out the GCF of -6y out of the second group


%28x-6y%29%28x-2y%29 Since we have a common term of x-2y, we can combine like terms

So 1x%5E2-2xy-6xy%2B12y%5E2 factors to %28x-6y%29%28x-2y%29


So this also means that 1x%5E2-8xy%2B12y%5E2 factors to %28x-6y%29%28x-2y%29 (since 1x%5E2-8xy%2B12y%5E2 is equivalent to 1x%5E2-2xy-6xy%2B12y%5E2)



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Answer:
So x%5E2-8xy%2B12y%5E2 factors to %28x-6y%29%28x-2y%29








# 4: Choose one factor of the following expression from the list below.
6x2 - 5x - 4




Looking at 6x%5E2-5x-4 we can see that the first term is 6x%5E2 and the last term is -4 where the coefficients are 6 and -4 respectively.

Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient -5? Let's list all of the factors of -24:



Factors of -24:
1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24
(1)*(-24)
(2)*(-12)
(3)*(-8)
(4)*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5

First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From this list we can see that 3 and -8 add up to -5 and multiply to -24


Now looking at the expression 6x%5E2-5x-4, replace -5x with 3x%2B-8x (notice 3x%2B-8x adds up to -5x. So it is equivalent to -5x)

6x%5E2%2Bhighlight%283x%2B-8x%29%2B-4


Now let's factor 6x%5E2%2B3x-8x-4 by grouping:


%286x%5E2%2B3x%29%2B%28-8x-4%29 Group like terms


3x%282x%2B1%29-4%282x%2B1%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of -4 out of the second group


%283x-4%29%282x%2B1%29 Since we have a common term of 2x%2B1, we can combine like terms

So 6x%5E2%2B3x-8x-4 factors to %283x-4%29%282x%2B1%29


So this also means that 6x%5E2-5x-4 factors to %283x-4%29%282x%2B1%29 (since 6x%5E2-5x-4 is equivalent to 6x%5E2%2B3x-8x-4)



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Answer:
So 6x%5E2-5x-4 factors to %283x-4%29%282x%2B1%29



So you could either choose 3x-4 or 2x%2B1