Question 27303: i want to help my child with this problem..please help me!!!
the product of two consecutive integers is 240. find the integers.
thanks a lot!!!
Answer by sdmmadam@yahoo.com(530)   (Show Source):
You can put this solution on YOUR website! the product of two consecutive integers is 240. find the integers.
What are two consecutive integers?
Two Consecutive integers are integers that differ by 1 in value.
Let the required consecutive integers be y and (y + 1)
Given that the product of y and (y + 1) is 240
That is y(y+1) = 240
y^2 + y = 240
y^2 + y - 240 = 0 Transposing(change side then change sign)
We now have a quadratic equation which is solved either by the squaring method or by factorization method.
Now Factorizing
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)
(y^2 + 16y) - 15y - 240 = 0
y(y+16)- 15(y+16) = 0
(y+16)(y-15) = 0
y+16 = 0 implies y = -16 which is negative and hence is not one of our numbers.
y-15 = 0 implies y = 15 and which further implies y+1 = 16
and of course 15X16 = 240
Answer: The consecutive numbers are 15 and 16
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.
Note: By the squaring method.
y^2 + y = 240
y^2 + 2(y)(1/2) + 1/4 = 240 +1/4
y^2 + 2(y)(1/2) + (1/2)^2 = 240 +1/4
(note that the LHS is of the form a + b)^2 with a= y and b = 1/2
(y + 1/2)^2 = 240 + 1/4
[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)]
(y + 1/2)^2 = (240X4 + 1)/4
(y + 1/2)^2 = 961/4
Taking the positive square root
(y + 1/2) = 31/2
y = 31/2 - 1/2 = (31-1)/2 = 30/2 =15
The rest is as above.
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