Problem 40 - College Algebra with Modeling and Visualization

Question

Solve each quadratic equation (a) graphically, (b) numerically, and (c) symbolically. Express graphical and numerical solutions to the nearest tenth when appropriate. $$ x^{2}-225=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 15\) and \(x = -15\).

Step 1: Rearrange the Equation

First, we note that we need to solve the quadratic equation \( x^2 - 225 = 0 \). The goal is to find values of \( x \) that make this equation true.

Step 2: Solve Graphically

We start by considering \( y = x^2 - 225 \) and graph this equation. The graph of \( y = x^2 - 225 \) is a parabola opening upwards. To find the solutions, we locate where the graph intersects the x-axis. These intersections give us the solutions to \( x^2 - 225 = 0 \). Here, the graph crosses the x-axis at \( x = 15 \) and \( x = -15 \).

Step 3: Solve Numerically

Since the equation is already in a simple form, we set up: \( x^2 = 225 \). Taking the square root of both sides, we find \( x = \pm \sqrt{225} \). Calculating, we get \( x = \pm 15 \). Thus, the numerical solutions to one decimal place are \( x = 15.0 \) and \( x = -15.0 \).

Step 4: Solve Symbolically

For symbolic solving, we rewrite \( x^2 - 225 = 0 \) as \( (x - 15)(x + 15) = 0 \) using the difference of squares formula. Then, we set each factor to zero: \( x - 15 = 0 \) and \( x + 15 = 0 \). Solving both equations gives \( x = 15 \) and \( x = -15 \).

Key Concepts

Graphical Solutions

To solve quadratic equations graphically, we convert the equation into a function of the form \( y = f(x) \). Taking our example equation \( x^2 - 225 = 0 \), we rewrite it as \( y = x^2 - 225 \). This equation represents a parabola that opens upwards. Graphically solving means identifying where this parabola intersects the x-axis. These intersection points are the solutions to the equation as they satisfy \( y = 0 \). Through graphing, one could observe the parabola intersecting the x-axis at points \( x = 15 \) and \( x = -15 \). Hence, these are the graphical solutions. Graphical solutions offer a visual perspective and allow us to understand the nature of the roots and solutions at a glance. Graphing can sometimes be less precise for very accurate solutions, especially if not using a digital tool or calculator. However, it provides an intuitive grasp of how the equation behaves.

Numerical Solutions

Numerical solutions involve using calculations to find the roots of the quadratic equation with the help of numerical methods or tools. In the equation \( x^2 - 225 = 0 \), rearranging gives \( x^2 = 225 \). We then proceed by taking the square root of both sides to find \( x = \pm \sqrt{225} \). This calculation yields \( x = 15 \) and \( x = -15 \). Since no additional decimals are required, the solutions remain simple to determine. Numerical approaches can include more complex methods such as the quadratic formula or using tools like spreadsheets or calculators to compute for equations needing more precision or when roots are decimals. The numerical method directly computes solutions, making it perfect for confirming results from other methods.

Symbolic Solutions

Symbolic solutions involve manipulating the equation algebraically to solve for \( x \) using known formulas or properties, such as factoring. In this example, we have \( x^2 - 225 = 0 \), which can be rewritten as a difference of squares: \( (x - 15)(x + 15) = 0 \). By setting each factor equal to zero: \( x - 15 = 0 \) and \( x + 15 = 0 \), we solve for \( x \) and find \( x = 15 \) and \( x = -15 \). This method utilizes well-known algebraic identities and properties, making it efficient for equations that can be easily factored. Symbolic solutions are valued in mathematics for their exactness and reliability, showing a clear logical process from start to finish for solving quadratic equations.