SOLUTION: The difference between two positive integers is 3. If the smaller is added to the square of the larger, the sum is 107. Find the integers.

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Question 1097048: The difference between two positive integers is 3. If the smaller is added to the square of the larger, the sum is 107. Find the integers.
Found 2 solutions by josh_jordan, MathTherapy:
Answer by josh_jordan(263)   (Show Source): You can put this solution on YOUR website!
We need to first set this up as two equations. We know that the difference between two positive integers is 3, So, our first equation would be

a - b = 3

We also know that if the smaller is added to the square of the larger, the sum is 107. So, our second equation would be



Now, we can rewrite equation 1 in terms of one of the variables. It can be either, so I'll choose to rewrite equation 1 in terms of b:

a - b = 3 =====> -b = 3 - a =====> b = -3 + a =====> b = a - 3

We can now replace b with a - 3 in equation 2:



Next, subtract 107 from both sides of the equation, giving us



We can then factor the polynomial on the left side of the equal sign. To save time, I'll just give you the polynomial in factored form and assume you know how to factor polynomials:

(a + 11)(a - 10)

Next, set each set of parenthesis equal to zero to find both values of a:

a + 11 = 0 =====> a = -11
a - 10 = 0 =====> a = 10

Our problem states that both integers are positive, so we can discard -11. This means 10 is one of our integers. We can now replace a with 10 in equation 1 to find our second positive integer:

10 - b = 3 =====> -b = -7 =====> b = 7

Our two integers are 10 and 7.

We can plug these amounts into equation 2 to verify:

=====> 107 = 107 (both sides of the equal sign match so our two integers are correct.

ANSWER: 10 & 7
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

The difference between two positive integers is 3. If the smaller is added to the square of the larger, the sum is 107. Find the integers.
Let smaller number be S

Then larger = S + 3

We then get: 





(S - 7)(S + 14) = 0

Smaller number or           OR         S = - 14 (ignore)

Larger number:  

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