document.write( "Question 1008465: One positive number is three more than twice a number that the product equals 1430.what are the numbers? \n" ); document.write( "

Algebra.Com's Answer #624157 by KMST(5328) $\"\"$ $\"About$

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THE UNEXPECTED WAY TO SOLVE IT:
\n" ); document.write( "The prime factorization of $\"1430\"$ is
\n" ); document.write( " $\"1430=2%2A5%2A11%2A13\"$ .
\n" ); document.write( "Because there are $\"4\"$ prime factors, each with an exponent of $\"green%281%29\"$ ,
\n" ); document.write( "the number of possible factors of $\"1430\"$ is
\n" ); document.write( "

.
\n" ); document.write( "That makes $\"16%2F2=8\"$ factor pairs:
\n" ); document.write( " $\"1%2A1430=1430\"$ ,
\n" ); document.write( " $\"2%2A715=1430\"$ ,
\n" ); document.write( " $\"5%2A286=1430\"$ ,
\n" ); document.write( " $\"10%2A143=1430\"$ ,
\n" ); document.write( " $\"11%2A130=1430\"$ ,
\n" ); document.write( " $\"13%2A110=1430\"$ ,
\n" ); document.write( " $\"22%2A65=1430\"$ , and
\n" ); document.write( " $\"26%2A55=1430\"$ .
\n" ); document.write( "The first six pairs have factors that are too far apart to be solutions.
\n" ); document.write( " $\"22%2A2%2B3=44%2B3=47%3C%3E65\"$ so $\"22\"$ and $\"65\"$ are not the two factors.
\n" ); document.write( " $\"26%2A2%2B3=52%2B3=55\"$ , so $\"highlight%28system%2826%2C%22and%22%2C55%29%29\"$ are the two numbers.
\n" ); document.write( "
\n" ); document.write( "THE EXPECTED WAY TO SOLVE IT:
\n" ); document.write( " $\"x\"$ = the smallest positive number mentioned first
\n" ); document.write( " $\"2x%2B3\"$ = the second number = three more than twice the first number
\n" ); document.write( " $\"x%282x%2B3%29=1430\"$ = the product of the two numbers
\n" ); document.write( " $\"x%282x%2B3%29=1430\"$ <--> $\"2x%5E2%2B3x=1430\"$ <--> $\"2x%5E2%2B3x-1430=0\"$
\n" ); document.write( "You can solve the quadratic equation above by the method of your choice
\n" ); document.write( "(\"completing the square\", factoring, or using the quadratic formula),
\n" ); document.write( "and you get the solutions $\"system%28x=26%2C%22and%22%2Cx=-55%2F2%29\"$ .
\n" ); document.write( "Factoring:
\n" ); document.write( " $\"2x%5E2%2B3x-1430=0\"$ --> $\"2x%5E2-52x%2B55x-1430=0\"$ --> $\"2x%28x-26%29%2B55%28x-26%29=0\"$ --> $\"%282x%2B55%29%28x-26%29=0\"$ --> $\"system%7Bx-26=0%2C%22or%22%2C2x%2B55=0%29\"$ --> $\"system%7Bx=26%2C%22or%22%2Cx=-55%2F2%29\"$
\n" ); document.write( "Since the first number mentioned was a positive number,
\n" ); document.write( "the solution is $\"highlight%28x=26%29\"$ ---> $\"2x%2B3=2%2A26%2B3\"$ ---> $\"highlight%282x%2B3=55%29\"$
\n" ); document.write( "Using the quadratic formula,
\n" ); document.write( "which says that the solutions to $\"ax%5E2%2Bbx%2Bc=0\"$
\n" ); document.write( "are given by $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"$ :
\n" ); document.write( "In this case $\"a=2\"$ , $\"b=3\"$ , ans $\"c=-1430\"$ , so
\n" ); document.write( " $\"x=%28-3+%2B-+sqrt%283%5E2-4%2A2%2A%28-1430%29%29%29%2F%282%2A2%29\"$
\n" ); document.write( " $\"x=%28-3+%2B-+sqrt%289%2B11440%29%29%2F4\"$
\n" ); document.write( " $\"x=%28-3+%2B-+sqrt%2811449%29%29%2F4\"$
\n" ); document.write( " $\"x=%28-3+%2B-+107%29%2F4\"$ ---> $\"system%28x=104%2F4=26%2C%22or%22%2Cx=-110%2F4=-55%2F2%29\"$
\n" ); document.write( "Completing the square:
\n" ); document.write( " $\"2x%5E2%2B3x=1430\"$
\n" ); document.write( " $\"x%5E2%2B%283%2F2%29x=715\"$
\n" ); document.write( " $\"x%5E2%2B%283%2F2%29x%2B%283%2F4%29%5E2-%283%2F4%29%5E2=715\"$
\n" ); document.write( " $\"%28x%2B3%2F4%29%5E2-9%2F16=715\"$
\n" ); document.write( " $\"%28x%2B3%2F4%29%5E2=715%2B9%2F16\"$
\n" ); document.write( " $\"%282x%2B3%2F4%29%5E2=11440%2F16%2B9%2F16\"$
\n" ); document.write( " $\"%28x%2B3%2F4%29%5E2=11449%2F16\"$
\n" ); document.write( " $\"%28x%2B3%2F4%29%5E2=%28107%2F4%29%5E2\"$ --> $\"system%28x%2B3%2F4=107%2F4%2C%22or%22%2Cx%2B3%2F4=-107%2F4%29\"$ --> $\"system%28x=107%2F4-3%2F4%2C%22or%22%2Cx%2B3%2F4=-107%2F4-3%2F4%29\"$ --> $\"system%28x=104%2F4=26%2C%22or%22%2Cx%2B3%2F4=-110%2F4=-55%2F2%29\"$ \n" ); document.write( "

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