SOLUTION: One positive number is three more than twice a number that the product equals 1430.what are the numbers?

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Question 1008465: One positive number is three more than twice a number that the product equals 1430.what are the numbers?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
THE UNEXPECTED WAY TO SOLVE IT:
The prime factorization of 1430 is
1430=2%2A5%2A11%2A13 .
Because there are 4 prime factors, each with an exponent of green%281%29 ,
the number of possible factors of 1430 is
.
That makes 16%2F2=8 factor pairs:
1%2A1430=1430 ,
2%2A715=1430 ,
5%2A286=1430 ,
10%2A143=1430 ,
11%2A130=1430 ,
13%2A110=1430 ,
22%2A65=1430 , and
26%2A55=1430 .
The first six pairs have factors that are too far apart to be solutions.
22%2A2%2B3=44%2B3=47%3C%3E65 so 22 and 65 are not the two factors.
26%2A2%2B3=52%2B3=55 , so highlight%28system%2826%2C%22and%22%2C55%29%29 are the two numbers.

THE EXPECTED WAY TO SOLVE IT:
x= the smallest positive number mentioned first
2x%2B3= the second number = three more than twice the first number
x%282x%2B3%29=1430= the product of the two numbers
x%282x%2B3%29=1430<-->2x%5E2%2B3x=1430<-->2x%5E2%2B3x-1430=0
You can solve the quadratic equation above by the method of your choice
("completing the square", factoring, or using the quadratic formula),
and you get the solutions system%28x=26%2C%22and%22%2Cx=-55%2F2%29 .
Factoring:
2x%5E2%2B3x-1430=0-->2x%5E2-52x%2B55x-1430=0-->2x%28x-26%29%2B55%28x-26%29=0-->%282x%2B55%29%28x-26%29=0-->system%7Bx-26=0%2C%22or%22%2C2x%2B55=0%29-->system%7Bx=26%2C%22or%22%2Cx=-55%2F2%29
Since the first number mentioned was a positive number,
the solution is highlight%28x=26%29--->2x%2B3=2%2A26%2B3--->highlight%282x%2B3=55%29
Using the quadratic formula,
which says that the solutions to ax%5E2%2Bbx%2Bc=0
are given by x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ :
In this case a=2 , b=3, ans c=-1430 , so
x=%28-3+%2B-+sqrt%283%5E2-4%2A2%2A%28-1430%29%29%29%2F%282%2A2%29
x=%28-3+%2B-+sqrt%289%2B11440%29%29%2F4
x=%28-3+%2B-+sqrt%2811449%29%29%2F4
x=%28-3+%2B-+107%29%2F4--->system%28x=104%2F4=26%2C%22or%22%2Cx=-110%2F4=-55%2F2%29
Completing the square:
2x%5E2%2B3x=1430
x%5E2%2B%283%2F2%29x=715
x%5E2%2B%283%2F2%29x%2B%283%2F4%29%5E2-%283%2F4%29%5E2=715
%28x%2B3%2F4%29%5E2-9%2F16=715
%28x%2B3%2F4%29%5E2=715%2B9%2F16
%282x%2B3%2F4%29%5E2=11440%2F16%2B9%2F16
%28x%2B3%2F4%29%5E2=11449%2F16
%28x%2B3%2F4%29%5E2=%28107%2F4%29%5E2-->system%28x%2B3%2F4=107%2F4%2C%22or%22%2Cx%2B3%2F4=-107%2F4%29-->system%28x=107%2F4-3%2F4%2C%22or%22%2Cx%2B3%2F4=-107%2F4-3%2F4%29-->system%28x=104%2F4=26%2C%22or%22%2Cx%2B3%2F4=-110%2F4=-55%2F2%29