Problem 56 - College Algebra

Question

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions. The squares of two numbers add to 360. The second number is half the value of the first number squared. What are the numbers?

Step-by-Step Solution

Answer

The numbers are \((6, 18)\) and \((-6, 18)\).

Step 1: Define Variables

Let's define the variables. Let \( x \) be the first number, and \( y \) be the second number. The problem provides relationships that can be expressed in terms of these variables.

Step 2: Set Up Equation from Sum of Squares

The problem states that the squares of two numbers add up to 360. Thus, the first equation is: \[ x^2 + y^2 = 360 \]

Step 3: Set Up Equation from Second Condition

The problem also tells us that the second number \( y \) is half the value of the first number squared. Thus, we can write the second equation as: \[ y = \frac{1}{2}x^2 \]

Step 4: Substitute Equation into First Equation

We can substitute \( y = \frac{1}{2}x^2 \) into the first equation. This gives us: \[ x^2 + \left( \frac{1}{2}x^2 \right)^2 = 360 \]

Step 5: Simplify the Equation

Simplify the expression: \[ x^2 + \frac{1}{4}x^4 = 360 \]

Step 6: Solve the Nonlinear Equation

To solve \( x^2 + \frac{1}{4}x^4 = 360 \), rearrange it to: \[ \frac{1}{4}x^4 + x^2 - 360 = 0 \] Let \( z = x^2 \), the equation becomes: \[ \frac{1}{4}z^2 + z - 360 = 0 \]. Solve using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).Here, \( a = \frac{1}{4} \), \( b = 1 \), \( c = -360 \).Calculate \( z \):\[ z = \frac{-1 \pm \sqrt{1^2 - 4 \cdot \frac{1}{4} \cdot (-360)}}{2 \cdot \frac{1}{4}} \]\[ z = \frac{-1 \pm \sqrt{1 + 360}}{\frac{1}{2}} \]\[ z = \frac{-1 \pm 19}{\frac{1}{2}} \]\[ z = 36, -20 \]Since \( z = x^2 \), and \( z = -20 \) gives no real solution, so \( x^2 = 36 \).

Step 7: Find Solutions for x and y

Solving \( x^2 = 36 \) gives us \( x = 6 \) or \( x = -6 \). Use \( y = \frac{1}{2}x^2\) to find \( y \).For \( x = 6 \):\( y = \frac{1}{2}(6)^2 = 18 \).For \( x = -6 \):\( y = \frac{1}{2}(-6)^2 = 18 \).Thus, the pairs are \((6, 18)\) and \((-6, 18)\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used for solving quadratic equations. A quadratic equation has the form

• \( ax^2 + bx + c = 0 \)

where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. Here's a breakdown of how the quadratic formula works:

• To solve any quadratic equation, the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is employed.
• This formula helps find the roots of the quadratic equation, where these 'roots' are the values of \( x \) that make the equation true.
• Critical to this is the discriminant, \( b^2 - 4ac \). The value under the square root (the discriminant) tells us about the nature of the roots:
  • If the discriminant is positive, there are two distinct real roots.
  • If it's zero, there's exactly one real root (a repeated root).
  • If negative, the roots are complex and not real.

By applying the quadratic formula, understanding different possible solutions becomes more manageable, as seen in solving equations related to various scientific and engineering problems. Always make sure to identify the coefficients \( a \), \( b \), and \( c \) correctly to avoid algebraic errors.

Systems of Equations

A system of equations is a collection of two or more equations with a common set of unknowns. In our exercise, we have a system of nonlinear equations since at least one equation includes variables to a power other than one:

• Such systems can help model real-world problems where several conditions or constraints are applicable simultaneously.
• The goal is often to find all possible solutions that satisfy all equations in the system.
• In our example, the problem was to find two numbers based on given conditions expressed in two equations:
  • \( x^2 + y^2 = 360 \) represents the sum of squares condition.
  • \( y = \frac{1}{2}x^2 \) represents the relationship between the numbers.
• In this approach, the solution involves manipulating equations simultaneously using a method like substitution or elimination.

Conceptually, solving systems of equations requires gathering useful insights from each equation to progress towards the final answer. It showcases how different relationships between variables can uniquely intersect to find a solution.

Substitution Method

The substitution method is an effective technique for solving systems of equations. Here's how it works for our problem, and generally:

• The goal is to solve one of the equations for one variable, and then substitute that expression into the other equation.

In our specific exercise, we used this method:

• First, we took the second condition \( y = \frac{1}{2}x^2 \) and substituted \( y \) in the first equation \( x^2 + y^2 = 360 \).
• This results in a single equation in terms of \( x \), which significantly simplifies the problem-solving process.
• After substitution, the original complex system became a standard equation
  • \( x^2 + \left( \frac{1}{2}x^2 \right)^2 = 360 \).

The substitution method can be especially handy when one of the equations is already solved for one variable. This simplifies problems significantly by decreasing the number of variables in the equation, allowing for a straightforward path to the solution.