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the product of two consecutive integers is 240. find the integers.\r
\n" ); document.write( "\n" ); document.write( "What are two consecutive integers?
\n" ); document.write( "Two Consecutive integers are integers that differ by 1 in value.
\n" ); document.write( "Let the required consecutive integers be y and (y + 1)
\n" ); document.write( "Given that the product of y and (y + 1) is 240
\n" ); document.write( "That is y(y+1) = 240
\n" ); document.write( "y^2 + y = 240
\n" ); document.write( "y^2 + y - 240 = 0 Transposing(change side then change sign)
\n" ); document.write( "We now have a quadratic equation which is solved either by the squaring method or by factorization method.
\n" ); document.write( "Now Factorizing
\n" ); document.write( "y^2 + (16y - 15y) - 240 = 0 (note that the product of the coefficient of y^2 term and the constant term is 1 X (-240) = -240 and the numerical two groups of all the factors of 240 are 16 and 15 whose difference is 1 and so the middle term (1)y is written as the difference between these two that is 16y - 15y, the larger number 16 being given the sign of the middle term)
\n" ); document.write( "(y^2 + 16y) - 15y - 240 = 0
\n" ); document.write( "y(y+16)- 15(y+16) = 0
\n" ); document.write( "(y+16)(y-15) = 0
\n" ); document.write( "y+16 = 0 implies y = -16 which is negative and hence is not one of our numbers.
\n" ); document.write( "y-15 = 0 implies y = 15 and which further implies y+1 = 16
\n" ); document.write( "and of course 15X16 = 240
\n" ); document.write( "Answer: The consecutive numbers are 15 and 16
\n" ); document.write( "Note: The negative value -16 does not fit into our context. It comes as one of the two values of the quadratic equation as a quadratic equation has always two values.
\n" ); document.write( "Note: By the squaring method.
\n" ); document.write( "y^2 + y = 240
\n" ); document.write( "y^2 + 2(y)(1/2) + 1/4 = 240 +1/4
\n" ); document.write( "y^2 + 2(y)(1/2) + (1/2)^2 = 240 +1/4
\n" ); document.write( "(note that the LHS is of the form a + b)^2 with a= y and b = 1/2
\n" ); document.write( "(y + 1/2)^2 = 240 + 1/4
\n" ); document.write( "[Expressing the LHS as a perfect square,in the form (a + b)^2 = a^2 +2ab + b^2 observe that the root of y^2 is y and since the linear term y (that is the term with power 1)is positive the LHS is going to be of the form (y + something)^2 and that something is (1/2) of that positive coefficient 1 of (+y) and when we expand (y+1/2)^2 we get one extra quantity 1/4 and hence adding 1/4 to both the sides we get the above][The explanation is (2 X y X what) is (1)y which gives answer for that what as (1/2)]
\n" ); document.write( "(y + 1/2)^2 = (240X4 + 1)/4
\n" ); document.write( "(y + 1/2)^2 = 961/4
\n" ); document.write( "Taking the positive square root
\n" ); document.write( "(y + 1/2) = 31/2
\n" ); document.write( "y = 31/2 - 1/2 = (31-1)/2 = 30/2 =15
\n" ); document.write( "The rest is as above.\r
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