Problem 54 - Brief Applied Calculus

Question

Solve each equation using a graphing calculator. [Hint: Begin with the window \([-10,10]\) by \([-10,10]\) or another of your choice (see Useful Hint in Graphing Calculator Terminology following the Preface) and use ZERO, SOLVE, or TRACE and ZOOM IN.] (Round answers to two decimal places.) $$ x^{2}+2 x-15=0 $$

Step-by-Step Solution

Answer

The solutions are approximately \(x = 3.00\) and \(x = -5.00\).

Step 1: Enter the Equation into the Calculator

Start by turning on your graphing calculator. Enter the equation \(y = x^2 + 2x - 15\) into the calculator's graphing function. This will plot the quadratic equation on the graph.

Step 2: Set the Graphing Window

Set the window for the graph. For starters, set the window range from \([-10, 10]\) for both the x-axis and the y-axis. This provides a broad view of where the parabola intersects the x-axis.

Step 3: Graph the Equation

After setting the window, press the graph button to display the parabola on the screen. The graph should show where the curve intersects the x-axis, which represents the roots of the equation.

Step 4: Find the Roots Using ZERO

Use the ZERO function in your calculator. This function helps identify the x-values where the graph crosses the x-axis. Move the cursor close to where the parabola intersects the x-axis and execute the ZERO function for both points.

Step 5: Record the Intersection Points

Once you've used the ZERO function, the calculator will display the x-coordinate of the intersection points. These are the solutions to the equation. Record each root, rounding to two decimal places if necessary.

Key Concepts

Quadratic Equation

A quadratic equation is simply an equation of the second degree. This means the highest exponent of the variable is 2. It generally takes the form \( ax^2 + bx + c = 0 \). In our specific problem, the quadratic equation is \( x^2 + 2x - 15 = 0 \). Quadratic equations often graph into a U-shaped curve called a parabola.
The parabola can either open upwards or downwards depending on the sign of the coefficient of \( x^2 \). In this equation, since the coefficient is positive (that's the 1 in front of \( x^2 \)), the parabola opens upward.
Key things to look out for in a quadratic equation include:

• **The Coefficients**: The values \( a, b, \) and \( c \), which in this case are 1, 2, and -15 respectively.
• **The Vertex**: The turning point of the parabola, which can be calculated using the formula \( h = -b/(2a) \).
• **The Axis of Symmetry**: A vertical line that runs through the vertex of the parabola. Its equation is the same \( x = -b/(2a) \).

Understanding these components makes it easier to graph and find solutions to quadratic equations.

Roots of Equation

The roots of an equation are the values of \( x \) where the function equals zero. In other words, they are the points where the graph of the equation intersects the x-axis. For a quadratic equation, these roots (also known as solutions or zeroes) can be found using a few different methods, one of which is graphing.
In our attempt to solve \( x^2 + 2x - 15 = 0 \), we are looking for the x-values where this equation crosses (or touches) the x-axis. These are the points that make \( y = 0 \) when you replace \( y \) in the quadratic equation.
Using a graphing calculator, you can:

Graphing Functions

Graphing functions, especially quadratic ones, is an important skill in mathematics. It helps visualize the behavior of equations. For our quadratic equation \( x^2 + 2x - 15 = 0 \), using the graph provides a powerful visual way to find its roots.
Here's how you use a graphing calculator to achieve this:

• **Enter the Equation**: Input the quadratic equation into the calculator's function feature, often found under a 'Y=' button.
• **Set the Viewing Window**: Adjust the axes range to ideally \([-10, 10]\) for both x and y, providing a clear view of where the function intersects the axes.
• **Graph the Function**: Once set, use the graph function to display the curve on the screen.
• **Utilize ZERO Function**: Move the cursor close to the point of intersection and execute the 'ZERO' function to find the precise point where the curve crosses the x-axis.

The graphical representation not only reveals the roots but also gives you valuable insights into the shape and opening direction of the parabola. This visualization aids in understanding the effect of changing coefficients within the quadratic equation.