One integer is 5 more than another. Their product is 126. Find the integers.

Question

One integer is 5 more than another. Their product is 126. Find the integers.

Answer 1

At a lower grade level I would solve this problem by looking at the prime factors (including the negative ones) of 126 and seeing which pair of them differ by 5. If you are in a higher grade level, though, you can solve this algebraically as follows:

 

First, let's give names to our integers. Let's call "one integer" X and call "another" Y.

 

Second, read out the first sentence and think of math expressions that mean the same thing:

   "one integer" = "X"

   "is" = "="

   "5 more than" = "5 +"

   "another" = "Y"

Put them together to get

   X = 5 + Y

 

Third, read out the second sentence and think of math expressions that mean the same thing:

   "their product" --> "their" is referring to X and Y and the product of X and Y is X*Y

   "is" = "="

   "126" = "126"

Put them together to get

   X*Y = 126

 

Fourth, note that from the two equations you have, you are told the value of X in terms of Y. This will help you solve the second equation if you use substitution:

   X = 5 + Y and X*Y = 126

   (5 + Y)*Y = 126

   5*Y + Y² = 126

   Y^2 + 5Y - 126 = 0

 

Now we have a quadratic equation. To solve for Y we need to use the quadratic formula:

   Y = [-5 + sqrt(52 - 4*1*(-126))]/2(1)

   Y = [-5 + sqrt(25 + 504)]/2

   Y = [-5 + sqrt(529)]/2

   Y = [-5 + 23]/2

   Y = [-5 + 23]/2 OR [-5 - 23]/2

   Y = 9 OR -14

 

If Y = 9, then X = 5 + 9 = 14.

If Y = -14, then X = 5 + (-14) = -9

 

So your numbers are either 9 and 14 or -9 and -14.

Answer 2

Let X be one integer.  The other integer is X + 5.  Since the two integers multiply together to give 126, we have:

 

X (X + 5) = 126

 

X² + 5X = 126 (Try completing the square. Take 1/2 of 5 in the 5X term.  This 5/2. Square this and add and subtract it to the left hand side of the equation.)

 

X² + 5X + 25/4 - 25/4 = 126  (Now rewrite X² + 5X +25/4 as an expression squared.)

 

(X + 5/2)² - 25/4 = 126  (Now add 25/4 to both sides.)

 

(X + 5/2)² = 25/4 + 126 (Make the right hand side into an improper fraction.)

 

(X + 5/2)² = 529/4 (Now take the square root of both sides.)

 

(X + 5/2) = ±23/2 (Now solve for X.)

 

X = 23/2 - 5/2 = 18/2 = 9  OR  X = -23/2 - 5/2 = -28/2 = -14

 

If X = 9, then X+5 = 14.  14 x 9 = 126 so this combination is one solution.

 

If X = -14 then X + 5 = -9.  -14 x -9 = 126 so this is also another solution.