Problem 44 - Intermediate Algebra

Question

Set up an equation and solve each problem. Suppose that the sum of two numbers is 20 , and the sum of their squares is 232 . Find the numbers.

Step-by-Step Solution

Answer

The numbers are 14 and 6.

Step 1: Define Variables

Let one number be \( x \) and the other number be \( y \).

Step 2: Set Up the First Equation

According to the problem, the sum of the two numbers is 20. This can be written as the equation \( x + y = 20 \).

Step 3: Set Up the Second Equation

The sum of the squares of the numbers is given as 232. This gives us the equation \( x^2 + y^2 = 232 \).

Step 4: Express One Variable in Terms of the Other

From the first equation, express \( y \) in terms of \( x \): \( y = 20 - x \).

Step 5: Substitute into the Second Equation

Substitute \( y = 20 - x \) into the second equation: \( x^2 + (20-x)^2 = 232 \).

Step 6: Simplify the Equation

Expand \((20-x)^2\) to get \(400 - 40x + x^2\), so the equation becomes \(x^2 + 400 - 40x + x^2 = 232\).

Step 7: Combine Like Terms

Combine the terms: \(2x^2 - 40x + 400 = 232\).

Step 8: Simplify Further

Subtract 232 from both sides to obtain \(2x^2 - 40x + 168 = 0\).

Step 9: Divide the Entire Equation by 2

Simplify the equation by dividing everything by 2: \(x^2 - 20x + 84 = 0\).

Step 10: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -20\), \(c = 84\).

Step 11: Simplify Using the Quadratic Formula

Calculate the discriminant: \((-20)^2 - 4 \times 1 \times 84 = 400 - 336 = 64\). Then solve for \( x \): \(x = \frac{20 \pm \sqrt{64}}{2}\).

Step 12: Calculate the Solutions for x

The solutions are \(x = \frac{20 + 8}{2} = 14\) and \(x = \frac{20 - 8}{2} = 6\).

Step 13: Find the Corresponding y Values

If \(x = 14\), then \(y = 20 - 14 = 6\). Conversely, if \(x = 6\), then \(y = 20 - 6 = 14\).

Step 14: Conclude the Solution

The numbers are 14 and 6.

Key Concepts

Quadratic Equations

Quadratic equations are a crucial part of algebra. They are used to model various real-world situations, such as projectile motion or optimizing profits. A quadratic equation in its standard form is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients. Here, \( a \) should not be zero, otherwise, it ceases to be quadratic.

Solving a quadratic equation means finding the values of \( x \) that make the equation true. Some common methods include:

• Factoring: This involves expressing the quadratic as a product of two binomials.
• Completing the Square: This converts the quadratic into a perfect square trinomial.
• Quadratic Formula: This is a universal method used when other methods are difficult. It is given as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Understanding these methods is vital for solving quadratic equations efficiently. In the provided exercise, we used the quadratic formula after simplifying the expression through several steps. The discriminant, \( b^2 - 4ac \), played a key role by telling us there are two possible real solutions since it was positive.

Systems of Equations

Systems of equations occur frequently in algebra. They involve finding values for variables that satisfy multiple equations at the same time. In the context of the exercise provided, we dealt with two variables, \( x \) and \( y \), through a system of equations.

When solving a system of equations, you can use several methods:

• Substitution: One variable is expressed in terms of another and substituted into the other equation, simplifying the solution process.
• Elimination: You eliminate one variable by adding or subtracting equations.
• Graphical Methods: Plotting the equations to find the point(s) where they intersect.

In the problem, we used the substitution method. This involved expressing \( y \) in terms of \( x \) from the first equation and substituting into the second equation. This enabled us to transform the system into a single quadratic equation, which we then solved.

Problem-Solving Strategies

Effective problem-solving in algebra often requires a strategic approach. It is crucial to break down the problem into manageable parts and tackle each systematically. Here are some useful strategies:

• Understand the Problem: Begin by carefully reading the problem to identify what is being asked.
• Define the Variables: Assign letters to unknown quantities, which simplifies manipulation of the equations.
• Set up Equations: Use the problem's conditions to express relationships in equation form.
• Choose a Solution Method: Decide on the best method for solving the equation based on the type and number of equations.
• Check Your Work: After finding solutions, substitute them back into the original equations to ensure they satisfy the conditions.

In our exercise, each step was crafted to adhere to these strategies. Starting with understanding variables, to setting up a system of equations, and eventually solving them, each move was methodical. The precision in solving and checking solutions ultimately leads to the correct answer.