Question 1008465
THE UNEXPECTED WAY TO SOLVE IT:
The prime factorization of {{{1430}}} is
{{{1430=2*5*11*13}}} .
Because there are {{{4}}} prime factors, each with an exponent of {{{green(1)}}} ,
the number of possible factors of {{{1430}}} is
{{{(green(1)+1)*(green(1)+1)*(green(1)+1)*(green(1)+1)=(green(1)+1)^4=2^4=16}}} .
That makes {{{16/2=8}}} factor pairs:
{{{1*1430=1430}}} ,
{{{2*715=1430}}} ,
{{{5*286=1430}}} ,
{{{10*143=1430}}} ,
{{{11*130=1430}}} ,
{{{13*110=1430}}} ,
{{{22*65=1430}}} , and
{{{26*55=1430}}} .
The first six pairs have factors that are too far apart to be solutions.
{{{22*2+3=44+3=47<>65}}} so {{{22}}} and {{{65}}} are not the two factors.
{{{26*2+3=52+3=55}}} , so {{{highlight(system(26,"and",55))}}} are the two numbers.
 
THE EXPECTED WAY TO SOLVE IT:
{{{x}}}= the smallest positive number mentioned first
{{{2x+3}}}= the second number = three more than twice the first number
{{{x(2x+3)=1430}}}= the product of the two numbers
{{{x(2x+3)=1430}}}<-->{{{2x^2+3x=1430}}}<-->{{{2x^2+3x-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(x=26,"and",x=-55/2)}}} .
Factoring:
{{{2x^2+3x-1430=0}}}-->{{{2x^2-52x+55x-1430=0}}}-->{{{2x(x-26)+55(x-26)=0}}}-->{{{(2x+55)(x-26)=0}}}-->{{{system{x-26=0,"or",2x+55=0)}}}-->{{{system{x=26,"or",x=-55/2)}}}
Since the first number mentioned was a positive number,
the solution is {{{highlight(x=26)}}}--->{{{2x+3=2*26+3}}}--->{{{highlight(2x+3=55)}}}
Using the quadratic formula,
which says that the solutions to {{{ax^2+bx+c=0}}}
are given by {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} :
In this case {{{a=2}}} , {{{b=3}}}, ans {{{c=-1430}}} , so
{{{x=(-3 +- sqrt(3^2-4*2*(-1430)))/(2*2)}}}
{{{x=(-3 +- sqrt(9+11440))/4}}}
{{{x=(-3 +- sqrt(11449))/4}}}
{{{x=(-3 +- 107)/4}}}--->{{{system(x=104/4=26,"or",x=-110/4=-55/2)}}}
Completing the square:
{{{2x^2+3x=1430}}}
{{{x^2+(3/2)x=715}}}
{{{x^2+(3/2)x+(3/4)^2-(3/4)^2=715}}}
{{{(x+3/4)^2-9/16=715}}}
{{{(x+3/4)^2=715+9/16}}}
{{{(2x+3/4)^2=11440/16+9/16}}}
{{{(x+3/4)^2=11449/16}}}
{{{(x+3/4)^2=(107/4)^2}}}-->{{{system(x+3/4=107/4,"or",x+3/4=-107/4)}}}-->{{{system(x=107/4-3/4,"or",x+3/4=-107/4-3/4)}}}-->{{{system(x=104/4=26,"or",x+3/4=-110/4=-55/2)}}}