SOLUTION: If the product of two consecutive even intergers is 34 larger than their sum, then what are the intergers?

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Question 148007: If the product of two consecutive even intergers is 34 larger than their sum, then what are the intergers?
Found 2 solutions by jim_thompson5910, ankor@dixie-net.com:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember, consecutive even integers follow the pattern: 2x, 2x%2B2, 2x%2B4, etc.


So if the "product of two consecutive even intergers is 34 larger than their sum", then we have the equation 2x%282x%2B2%29=%282x%29%2B%282x%2B2%29%2B34

2x%282x%2B2%29=2x%2B2x%2B2%2B34 Start with the given equation.


4x%5E2%2B4x=2x%2B2x%2B2%2B34 Distribute.


4x%5E2%2B4x=4x%2B36 Combine like terms


4x%5E2%2B4x-4x-36=0 Subtract 4x from both sides. Subtract 36 from both sides.


4x%5E2%2B0x-36=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=4, b=0, and c=-36


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-%280%29+%2B-+sqrt%28+%280%29%5E2-4%284%29%28-36%29+%29%29%2F%282%284%29%29 Plug in a=4, b=0, and c=-36


x+=+%28-0+%2B-+sqrt%28+0-4%284%29%28-36%29+%29%29%2F%282%284%29%29 Square 0 to get 0.


x+=+%28-0+%2B-+sqrt%28+0--576+%29%29%2F%282%284%29%29 Multiply 4%284%29%28-36%29 to get -576


x+=+%28-0+%2B-+sqrt%28+0%2B576+%29%29%2F%282%284%29%29 Rewrite sqrt%280--576%29 as sqrt%280%2B576%29


x+=+%28-0+%2B-+sqrt%28+576+%29%29%2F%282%284%29%29 Add 0 to 576 to get 576


x+=+%28-0+%2B-+sqrt%28+576+%29%29%2F%288%29 Multiply 2 and 4 to get 8.


x+=+%28-0+%2B-+24%29%2F%288%29 Take the square root of 576 to get 24.


x+=+%28-0+%2B+24%29%2F%288%29 or x+=+%28-0+-+24%29%2F%288%29 Break up the expression.


x+=+%2824%29%2F%288%29 or x+=++%28-24%29%2F%288%29 Combine like terms.


x+=+3 or x+=+-3 Simplify.




So the first number is either 2%283%29=6 or 2%28-3%29=-6

And the second number is either 2%283%29%2B2=8 or 2%28-3%29%2B2=-4

This means that the two numbers are 6 and 8

OR

the two numbers are -6 and -4


Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
If the product of two consecutive even integers is 34 larger than their sum, then what are the integers?
;
x(x+2) = x + (x+2) + 34
:
x^2 + 2x = 2x + 36
:
x^2 + 2x - 2x - 36 = 0
:
x^2 - 36 = 0
Factors to:
(x-6)(x+6) = 0
:
x = +6
and
x = -6
:
integers are; 6,8 and -6,-4