If two consecutive positive integers have the property that one integer times twice the other equals 612 what is the sum of these two integers?

Answer 1
If the smaller integer is #x# we have either #(x) xx 2(x+1) = 612# or #2(x) xx (x+1) = 612#
in either case #color(white)("XXXXX")##2(x)(x+1) =612#
#color(white)("XXXXX")#2x^2+2x = 612#
#color(white)("XXXXX")#2x^2+2x-612 = 0#
Using the quadratic formula #color(white)("XXXXX")##x = (-2+-sqrt(4+4(2)(612)))/(2(2)#
#color(white)("XXXXX")##=(-2+-sqrt(4900))/4#
#color(white)("XXXXX")##=(-2+-70)/4# (ignoring the negative value, since we are told the numbers are positive) #color(white)("XXXXX")##x=68/4 = 17#

and the two consecutive positive integers are 17 and 18

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Answer 2

Let ( x ) be the first positive integer. Then, the next consecutive positive integer is ( x + 1 ). According to the given condition, we have the equation ( x(2(x + 1)) = 612 ).

Expanding and simplifying:

( x(2x + 2) = 612 )

( 2x^2 + 2x = 612 )

Divide both sides by 2:

( x^2 + x = 306 )

Rearrange the equation:

( x^2 + x - 306 = 0 )

Now, we can use the quadratic formula to solve for ( x ):

( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )

Where ( a = 1 ), ( b = 1 ), and ( c = -306 ).

( x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-306)}}{2(1)} )

( x = \frac{-1 \pm \sqrt{1 + 1224}}{2} )

( x = \frac{-1 \pm \sqrt{1225}}{2} )

( x = \frac{-1 \pm 35}{2} )

Since ( x ) must be positive, we take the positive root:

( x = \frac{-1 + 35}{2} = \frac{34}{2} = 17 )

So, the first positive integer is 17, and the next consecutive positive integer is ( 17 + 1 = 18 ).

Therefore, the sum of these two integers is ( 17 + 18 = 35 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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