Integers: Finding the Difference from Sum and Product

This problem asks us to find the difference between two integers when their sum and product are known. We can approach this using algebraic identities or by forming and solving a quadratic equation.

Understanding the Integer Problem Statement

We are given two pieces of information about two unknown integers:

• The sum of the two integers is 26.
• The product of the two integers is 165.

Our goal is to determine the absolute difference between these two integers.

Method 1: Using Algebraic Identity to Find Difference

Let the two integers be \(x\) and \(y\). Based on the problem statement, we can write down two equations:

1. Sum of integers: \(x + y = 26\)
2. Product of integers: \(xy = 165\)

We need to find the value of \(|x - y|\). A useful algebraic identity connects the sum, product, and difference of two numbers:

\((x - y)^2 = (x + y)^2 - 4xy\)

Now, let's substitute the given values from our equations into this identity:

• First, calculate \((x + y)^2\):
• \((26)^2 = 26 \times 26 = 676\)
• Next, calculate \(4xy\):
• \(4 \times 165 = 660\)
• Substitute these values back into the identity:
• \((x - y)^2 = 676 - 660\)
• \((x - y)^2 = 16\)

To find \(x - y\), we take the square root of both sides:

\(x - y = \pm\sqrt{16}\)

\(x - y = \pm 4\)

The difference between the two integers, regardless of which one is larger, is the absolute value of \(x - y\). Therefore, the difference is \(|4|\) or \(|-4|\), which is 4.

Method 2: Solving by Forming a Quadratic Equation

Another way to solve this problem is to find the actual values of the two integers first. We have:

• \(x + y = 26\) (Equation 1)
• \(xy = 165\) (Equation 2)

From Equation 1, we can express \(y\) in terms of \(x\):

\(y = 26 - x\)

Substitute this expression for \(y\) into Equation 2:

\(x(26 - x) = 165\)

Expand the equation:

\(26x - x^2 = 165\)

Rearrange it into a standard quadratic equation form (\(ax^2 + bx + c = 0\)):

\(x^2 - 26x + 165 = 0\)

Now, we can solve this quadratic equation by factoring. We need two numbers that multiply to 165 and add up to -26. These numbers are -11 and -15.

So, the equation can be factored as:

\((x - 11)(x - 15) = 0\)

This gives us two possible values for \(x\):

• \(x - 11 = 0 \implies x = 11\)
• \(x - 15 = 0 \implies x = 15\)

If \(x = 11\), then using \(y = 26 - x\), we get \(y = 26 - 11 = 15\).

If \(x = 15\), then using \(y = 26 - x\), we get \(y = 26 - 15 = 11\).

In both cases, the two integers are 11 and 15.

Finally, we find the difference between these two integers:

Difference = \(|15 - 11| = 4\)

Both methods confirm that the difference between the two integers is 4.

Conclusion on Integer Difference

The difference between the two integers, whose sum is 26 and product is 165, is 4.