the product of two consecutive intergers is 6 less than 3 times the sum

Question

the product of two consecutive intergers is 6 less than 3 times the sum

Answer 1

Can you check the question ????

 

 

Answer 2

You Stephen meant to say the product of two consecutive EVEN or ODD integers is 6 less than 3 times their sum.

 

Here's why....

 

As written, CONSECUTIVE integers use the model x and x+1 to represent the first and second numbers respectively.

Then

 

x(x+1)+6 = 3(x + x+1)

 

x(x+1)+6 = 3(2x+1)

 

x^2 +x + 6 = 6x + 3

 

x^2 +x + 6 - 6x - 3 = 0

 

x^2 - 5x + 3 = 0

 

[5 +or- sqrt(  (-5)^2 - 4*1*3) ] / (2*1) =

 

[ 5 +or- sqrt( 25 - 12)]/2 =

 

[ 5 +or- sqrt(13)]/2

 

These are not integers!!!

 

Checking the result,

for real numbers [5 + sqrt(13)]/2 and the next "consecutive" number that is 1 greater, it is [7+ sqrt(13)]/2

 

The product of them is:

[5 + sqrt(13)/2][7 + sqrt(13)]/2 = 35 + 12*Sqrt(13) + 13]/4 = [48 + 12*sqrt(13)]/4 = 12 + 3*sqrt(13)

 

The sum of them is [12 + 2*sqrt(13)]/2 = 6 + sqrt(13)

 

So 3 times their sum less 6 is:

3 [ 6 + sqrt(13)]-6 = 18 + 3*Sqrt(13) - 6 = 12 + 3*sqrt(13)

 

So yes it checks 

 

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Now, if you had said consecutive even or "consecutive odd" then

the equation becomes:

 

 

x(x+2)+6 = 3(x+x+2)

x^2 + 2x + 6 = 3(2x+2)

x^2 + 2x + 6 = 6x + 6

x^2 - 4x = 0

 

x(x-4) = 0

x=0 x=4

 

For integers 0 and 2:

0*2 = 0 = 6 - 6 = 3*2 - 6 = 3(0+2)-6

 

AND

 

For integers

4 and 6:

4*6 = 24 = 30 - 6 = 3*10 - 6 = 3( 4+6) - 6

 

 

which better fits the description

 

please repost