product of two consecutive odd integers is 1 less than 4 times their sum. Find the two integers.

Question

The product of two consecutive odd integers is 1 less than 4 times their sum. Find the two integers. Answer in the form of paired points with the lowest of the two integers first.

Answer 1

Consecutive numbers mean that they follow each other.

 

2, 3, 4, 5, ... are consecutive numbers.

3, 5, 7, 9, ... are consecutive odd numbers.

 

If you look closely, adding 2 to an odd number gives you the next consecutive odd number.

If n = 1st number, then n + 2 = 2nd number.

 

Product means multiply, sum means add, and less than means subtract from.  Also, is always means equals.

 

Therefore,

n(n+2) = 4[n + (n+2)] - 1

n(n+2) = 4(2n + 2) -1

n²+2n = 8n + 8 -1

n²+2n = 8n + 7

n²-6n-7 =0

 

If you factor the quadratic:

(n-7)(n+1)=0

 

n-7 = 0 or n+1= 0

 

n=7  meaning the numbers are 7 and 9.

 

or 

n=-1 meaning the numbers are -1 and 1.

Answer 2

(x)(x+2)=4[(x)+(x+2)]-1

x^2+2x=4[2x+2]-1

x^2+2x=8x+8-1

x^2+2x=8x+7

x^2+2x-8x-7=0

x^2-6x-7=0

(x-7)(x+1)=0

x-7=0

x=7

x+2=9

one solution is 7 and 9

 

x+1=0

x=-1

x+2=1

a second solution is -1 and 1