plz help how do i solve

Question

The product of two consecutive odd integers is
57 less than

9 times their sum. Find the two integers. Answer in the form of paired points with the lowest of the two integers first

Answer 1

Hi Hayley,

 

To get an odd integer, you just need to take an even integer and add 1 to it (or subtract 1 from it).

 

Any integer multiplied by 2 will be even, so if we let x = any integer,

 

then 2x is an even integer, and 2x+1 is an odd integer.

 

So the 1st odd integer is 2x+1.

 

The consecutive odd integer is 2 up from the first, so the 2nd odd integer is 2x+1 +2 or

 

 

the 2nd odd integer is 2x+3

 

 

So the product of two consecutive odd integers is

 

(2x+1)(2x+3)

 

"... is 57 less than 9 times their sum"  

would translate to 

 

= (9) * [(2x+1) + (2x+3)] - 57

 

Putting all of this together, you get

 

(2x+1)(2x+3)=9(2x+1+2x+3)- 57

 

 

So FOIL the left and combine like terms inside the parentheses on the right and end up with

 

4x² + 8x + 3 = 9(4x + 4) - 57

 

Now distribute the 9 into the "( )"s

 

4x² + 8x + 3 = 36x + 36 - 57

 

4x² + 8x + 3 = 36x - 21

 

Now put everything on the left side (subract the 36x from both sides and add 21 to both sides) and get

 

4x² - 28x + 24 = 0

 

You can now divide a 4 out of every term, and get

 

x² - 7x + 6 = 0

 

And this factors into

 

(x-1)(x-6) = 0

 

So x = 1 or x = 6

 

But we are not looking for x, we are looking for (2x+1) and (2x+3),

so put in x=1 and get that one of the odd integers is 3, and the other is 5.

 

Now check:  Sum = 8, 9 times the sum = 72, 57 less than 72 is 15, and is 15 the product of these two consecutive odd integers?  Answer is :  YES!

 

So 3 and 5 works  as should  2(6)+1 =13 and 2(6)+3=15.

 

Let's check this last pair:

 

sum = 28, 9*sum = 9*28=252, 57 less than 252 is 195.

Is the product of 13 and 15 = 195?  YES!  So our second pair of consecutive odd integers is

 

13 and 15, or in the notation requested would be 

 

(3,5)  and (13, 15)