Problem 46 - College Algebra with Modeling and Visualization

Question

Solve the quadratic equation graphically. $$ x(x+24)=6912 $$

Step-by-Step Solution

Answer

Plot the function and identify x-intercepts.

Step 1: Simplify the Equation

The given equation is \( x(x+24)=6912 \). First, expand and simplify it to get a standard quadratic form: \( x^2 + 24x = 6912 \).

Step 2: Set to Zero

Subtract 6912 from both sides to transform the equation into standard quadratic form: \( x^2 + 24x - 6912 = 0 \).

Step 3: Graph the Quadratic Function

Plot the quadratic function \( f(x) = x^2 + 24x - 6912 \) on a coordinate plane. Use graphing software or a calculator capable of graphing.

Step 4: Identify the Roots

The roots of the quadratic equation correspond to the values of \( x \) where the graph intersects the x-axis. These are the solutions to the original equation.

Key Concepts

Graphical Solution

When solving a quadratic equation graphically, the focus is on translating the equation into a visual format. This process involves drawing the curve of the quadratic function on a graph. For the equation \( x(x + 24) = 6912 \), you first need to convert it into the standard quadratic form: \( x^2 + 24x - 6912 = 0 \).

This form is helpful because you can easily plot it using graphing software or a calculator. By graphing \( f(x) = x^2 + 24x - 6912 \), you create a visual representation of the equation. The curve of this graph is a parabola—typically U-shaped or upside down, depending on the sign of the quadratic term. In this case, it's a standard U-shape, as the coefficient of \( x^2 \) is positive.

Viewing a problem graphically aids comprehension, allowing you to see exactly where the function meets the x-axis. These intersection points are crucial, as they represent the solutions or roots of the equation. This method not only confirms calculations but also provides a visual context, making the abstract math more tangible.

Coordinate Plane

The coordinate plane is a two-dimensional space where we graph functions and equations such as our quadratic equation \( x^2 + 24x - 6912 \). It's composed of two axes: the horizontal x-axis and the vertical y-axis. Each point on the plane can be identified by an ordered pair \((x, y)\).

When plotting the quadratic function, you'll identify various points that satisfy \( f(x) = x^2 + 24x - 6912 \) and mark them on the plane. The collection of these points forms the parabola, providing a complete graphical picture.

Using a coordinate plane effectively highlights critical information such as the parabola's vertex, axis of symmetry, and, most importantly for this task, the roots where the parabola intersects the x-axis.

A clear understanding of the coordinate plane is fundamental, as it serves as the canvas for our graphical solutions and allows the examination of functional behavior and intersections analytically.

Roots of Equation

The roots of a quadratic equation are the values of \( x \) that make the equation equal to zero. For the quadratic equation \( x^2 + 24x - 6912 = 0 \), these roots can be seen as the points where the graph of the function \( f(x) = x^2 + 24x - 6912 \) intersects the x-axis.

These intersection points are significant because they are solutions to the equation. You identify them on the graph by looking for where the parabola crosses the horizontal axis, which will happen at one or two points for quadratic equations.

Understanding roots graphically adds another layer to solving equations, emphasizing not just the numerical answer but the geometric one as well. It's like solving the equation twice—once with numbers and once visually—reinforcing your comprehension of the solution's nature and its implications.