SOLUTION: Find two consecutive odd integers such that the sum of their reciprocals is 12/35.I have been trying for three days to write a problem out for this question and I am for some reaso

Question 294753: Find two consecutive odd integers such that the sum of their reciprocals is 12/35.I have been trying for three days to write a problem out for this question and I am for some reason not able to come up with an answer. Thank you
Found 2 solutions by richwmiller, jim_thompson5910:
Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
So where is your work?
Let x be first odd integer
x+2 be the second odd integer
What do we know?
The reciprocals add up to 12/35
1/x+1/(x+2)=12/35
We have one equation with one unknown
We get two answers
-7/6 which isn't an odd number and 5
check
1/5+1/7=12/35
7/35+5/35=12/35
ok
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Consecutive odd integers follow the pattern: x, x+2, ...

So adding their reciprocals to get means that

Start with the given equation

Multiply EVERY term by the LCD to clear out the fractions.

Distribute

Combine like terms.

Get every term to one side.

Combine like terms.

Solved by pluggable solver: Quadratic Formula

Let's use the quadratic formula to solve for x:

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=12, b=-46, and c=-70

Negate -46 to get 46

Square -46 to get 2116 (note: remember when you square -46, you must square the negative as well. This is because .)

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root (note: If you need help with simplifying the square root, check out this solver)

Multiply 2 and 12 to get 24

So now the expression breaks down into two parts

or

Lets look at the first part:

Add the terms in the numerator

Divide

So one answer is

Now lets look at the second part:

Subtract the terms in the numerator

Divide

So another answer is

So our solutions are:

or

Since we're only looking for integer values of 'x', we can ignore .

So the only answer is which means that the two consecutive odd integers are 5 and 7.

Notice how which confirms our answer.
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