Problem 18 - Beginning and Intermediate Algebra

Question

Solve. Find two integers whose sum is 26 and whose product is a maximum.

Step-by-Step Solution

Answer

The two integers that have a maximum product are both 13, with a product of 169.

Step 1: Define the variables

Let x and y be the two integers we are looking for. We are given that their sum is 26, which can be written as: \(x+y=26\)

Step 2: Express y in terms of x

We can solve the equation above for y in terms of x to get: \(y=26-x\)

Step 3: Define the product function

Let's define a function P(x) representing the product of x and y: \(P(x)=x(26-x)=26x-x^2\)

Step 4: Find the maximum value of the product function

We need to find the maximum value of the function P(x). For this, we need to find the vertex of the parabola \(P(x)=26x-x^2\). The x-coordinate of the vertex is given by the formula \(\frac{-b}{2a}\), where a and b are the coefficients of the quadratic equation. In this case, a = -1 and b = 26. So, the x-coordinate of the vertex is: \(x_v=\frac{-26}{2(-1)}=13\)

Step 5: Find the corresponding y-coordinate

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting x=13 into the equation we found in Step 2: \(y=26-13=13\)

Step 6: Verify the integers

Now we have two integers, x=13 and y=13, whose sum is 26 and whose product is maximum. The product of these integers is: \(P(13)=(13)(13)=169\) Thus, the two integers are 13 and 13 with a maximum product of 169.

Key Concepts

Integer Solutions

In many mathematical problems, finding integer solutions is key. Integers are numbers without decimal or fractional parts, such as -3, 0, 1, and so forth. In our problem, we need two integers that together add up to 26. To find these, we defined two variables, denoted as \(x\) and \(y\). One way to approach this is by using simple algebra. We write the equation \(x + y = 26\). By expressing one variable in terms of the other, we reduce the number of variables, simplifying the solution process.In this exercise, we expressed \(y\) in terms of \(x\), giving us the equation \(y = 26 - x\). This method allows us to focus on one variable at a time, while keeping our equations balanced and manageable. This type of methodical approach ensures that we maintain only integer values for our solutions. Working with integer solutions is crucial in number theory and helps in maintaining exact, rather than approximate, answers in problems resembling this product maximization challenge.

Product Maximization

When we're tasked with maximizing a product, especially under certain constraints like a fixed sum, quadratic optimization comes into play. Product maximization involves finding a point where multiplying two values yields a maximum result. In simpler terms, we're looking for the highest possible outcome from multiplying our two integers.Here's how we structured this for our specific problem:

• We first set up the product function \(P(x) = x(26-x) = 26x - x^2\). This quadratic equation depends on \(x\), as \(y\) is represented through \(x\).


• The shape of the function, which is a parabola, opens downward because the coefficient of \(x^2\) is negative (\(-1\)).


• This means the maximum product value occurs at the vertex of the parabola.

In maximizing the product, we don't just choose any two numbers whose sum is 26. We have to make sure that we apply quadratic optimization techniques to ensure the multiplication yields the highest possible figure. This understanding of conditions applied to maximize a product is crucial in solving similar math problems.

Vertex Formula

The vertex formula is a pivotal part of finding maximum or minimum values in quadratic equations. The vertex of a parabola \(ax^2 + bx + c\) is located at \(x = \frac{-b}{2a}\) and determines the turning point where the equation reaches its highest or lowest value.In our problem, we apply the vertex formula to \(P(x) = 26x - x^2\). Here:

• \(a = -1\) (since it's \(-x^2\)),


• \(b = 26\).

Using the formula, we found\[x_v = \frac{-26}{2(-1)} = 13\]This calculation gives us the \(x\)-coordinate of the vertex, and hence the value where the product function reaches its maximum. By substituting \(x = 13\) back into our expression for \(y\), we found \(y = 13\).The vertex formula is a helpful calculation tool. It allows you to pinpoint that crucial highest or lowest value that a quadratic function can achieve. In maximizing or minimizing such functions, especially in real-world applications or optimization problems, understanding and utilizing this formula is essential.