SOLUTION: The product of two consecutive positive even numbers is 288. By forming and solving an equation, find the larger of the two numbers

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Question 1059845: The product of two consecutive positive even numbers is 288. By forming and solving an equation, find the larger of the two numbers
Found 3 solutions by acw1213, Alan3354, MathTherapy:
Answer by acw1213(28) About Me  (Show Source):
You can put this solution on YOUR website!
Consecutive integers follow one another in an order.
If "2x" is a consecutive integer, it would be 2x, 2x + 1, 2x + 2... etc.
Let's say 2x will represent one of the numbers and 2x + 2 will represent another number.
Multiply them together
2x(2x + 2) = 288
Distribute
4x^2 + 4x = 288
We need to get this equal to zero to be able to solve.
Subtract 288 on both sides.
4x^2 - 4x - 288 = 0
We cannot factor this, so we must use the quadratic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
In our quadratic equation,
a = 4
b = -4
c = -288
Plug in "a" "b" and "c" and simplify.
x+=+%284+%2B-+sqrt%28+-4%5E2-4%2A4%2A-288+%29%29%2F%282%2A4%29+
Let's simplify the numerator in our square root fraction first.
-4%5E2 is 16.
-4(4)(-288) is 4608.
Now we have:
x+=+%284+%2B-+sqrt%2816+%2B+4608%29%29%2F%282%2A4%29+
16 + 4608 is 4624
x+=+%284+%2B-+sqrt%284624%29%29%2F%282%2A4%29+
Onto the denominator, 2*4 we know is 8.
x+=+%284+%2B-+sqrt%284624%29%29%2F%288%29+
We're not done just yet!
We do not know if 4 will be ADDED or SUBTRACTED from the square root of 578 (all over 8), so we have to solve for "x" using BOTH WAYS!
Let's begin with adding.
x+=+%28-4+%2B+sqrt%284624%29%29%2F8+
The square root of 4624 is 68.
x+=+%284+%2B+68%29%29%2F8+
4 + 68 is 72.
x+=+%2872%29%29%2F8+
And finally, 72 divided by 8 is 9.
We have one answer!
x+=+9
Now let's solve with SUBTRACTION
x+=+%284+-+68%29%29%2F8+
4 - 68 is -64.

x+=+%28-64%29%29%2F8+
-64 divided by 8 is -8.
Our other answer is
x+=+-8

Our solutions are x = -8 or x = 9.

We still need to find two possible pairs of numbers,
so plug in -8 for "x" in 2x to find one number, and plug in -8 for "x" in 2x + 2 and find that number.

2%2A%28-8%29+=+-16
We have one number, -16.
Now, plug in -8 for "x" in 2x + 2 to find the other number.
2%2A%28-8%29+%2B+2+=+-14%29
We have two numbers in this pair, -16 and -14.
Now, plug in 9 into the same two expressions to get two MORE possible solutions.
2%2A%28-9%29+=+-18
2%2A%28-9%29+%2B+2+=+-16

We have a total of 4 numbers now.
-16, -14
and
-18, -16.
Find the larger of both sets.


Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The product of two consecutive positive even numbers is 288. By forming and solving an equation, find the larger of the two numbers
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Find a pair of factors that differ by 2: --> 16 & 18 and -18 & -16
We're done.
===========================
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The long way around:
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x*(x+2) = 288
x^2 + 2x - 288 = 0
Now, find a pair of factors that differ by 2, same as above.
(x-16)*(x+18) = 0
etc.
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Extra steps and time for no good reason.
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The other tutor is trying to get a Master's degree from this problem.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

The product of two consecutive positive even numbers is 288. By forming and solving an equation, find the larger of the two numbers
Let the larger number be L

Then the smaller is: L - 2

We then get: L(L - 2) = 288________L%5E2+-+2L+-+288+=+0

Since the difference in the 2 numbers is 2, then find the square root of their product, or the square root of 288: sqrt%28288%29 = 16.97.

This means that one of the number is 16. Divide 288 by 16 and you get 18. The 2 numbers are 16 and 18. 

Now, L%5E2+-+2L+-+288+=+0 becomes: (L - 18)(L + 16) = 0 

Thus, the larger number can be 18 or - 16, but since you need the LARGER POSITIVE EVEN number, you have it. Done!!