The product of 2 concutive positive integers is 14 more than their sum. Set up an equation that can be used to find the 2 numbers and solve it.

Question

The product of 2 concutive positive integers is 14 more than their sum. Set up an equation that can be used to find the 2 numbers and solve it.

Answer 1

There may be something wrong with this problem statement. Neither X nor Y solves as an integer:

 

We're told that we have two "concutive" positive integers. I believe this is a typo and it really should be "consecutive". So if we call one of the integers X and the next consecutive positive integer Y, then Y=X+1.

 

Y=X+1

 

We are told that their product is 14 more than their sum. So, XY=X+Y+14

 

XY=X+Y+14

 

We can substitute for Y in terms of X in the second equation using the first equation.

 

XY=X(X+1) on the left side and

 

X+(X+1)+14 on the right side, so,

 

X(X+1)=X+(X+1)+14

 

Multiplying and gathering terms

 

X² +X=2X+1+14

 

and then simplifying

 

X² +X=2X+15

 

then putting into standard form for quadratic equations, which this IS since there is a variable squared, X² ,  but no higher powers

 

X² +X-2X-15=0 That is, finally,

 

X² -X-15 = 0

 

This corresponds to the standard quadratic form aX² + bX + c = 0

with

a = 1

b = -1

c = -15

 

The quadratic equation for the roots of this equation, for the values that solve it, is

 

X = [-b ± √b²-4ac]/2a (Check my transcription so you understand the symbology: the square root is of the entire term b2-4ac)

 

so X = [-(-1)±√((-1)² - 4(1)(-15))]/2(1)

 

or

 

X = [1±√(1+60)]/2 = (1±√61)/2

 

but

 

√61 = 7.81,

 

rounded to two decimal places.

 

We have

 

1±7.81

 

so if X is a positive integer as stated originally, we must ADD that to 1, because subtracting would give a negative quantity.

But

 

1+7.81 = 8.81

 

and dividing by 2 (the 2a divisor term in the quadratic formula) gives

 

X = 4.405 which would make 

 

Y = X+1 = 5.405

 

Which are NOT integers.

 

Checking the math

 

XY = 4.405 x 5.405 = 23.81,

 

rounded to two decimal places

 

X + Y + 14 = 4.405 + 5.405 +14 = 23.81

 

It checks, we solved the equations correctly, Y=X+1 and XY = X+Y+14, but neither X nor Y is an integer.

 

???

 

It seems something is wrong with the problem statement.