Problem 57 - 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.) $$ 4 x^{2}+24 x+45=9 $$

Step-by-Step Solution

Answer

The solutions rounded to two decimal places are approximately \( x_1 = -3.00 \) and \( x_2 = -3.00 \).

Step 1: Set Up the Equation

First, set the given equation to zero by subtracting 9 from both sides. This results in the equation: \[ 4x^2 + 24x + 36 = 0 \] This will allow us to find the x-intercepts, which represent the solutions to this equation.

Step 2: Enter the Equation in the Graphing Calculator

Enter the equation \( y = 4x^2 + 24x + 36 \) into the graphing calculator. Set the viewing window to \([-10, 10] \) for both x and y axes. Ensure that the quadratic graph is visible within this window.

Step 3: Identify Solutions Using ZERO Function

Graph the function and use the "ZERO" function on your calculator. Select the regions where the graph crosses the x-axis by setting bounds around it to find the roots accurately.

Step 4: Interpret the Calculator Results

The calculator will display the x-intercepts, or zeros, of the function. These values are your solutions to the equation. Round these values to two decimal places.

Key Concepts

X-intercepts

When solving quadratic equations using a graphing calculator, one of the key points you'll come across is finding the x-intercepts. These are the points where the graph of the equation crosses the x-axis. In our given equation, we adjusted it to the form \[4x^2 + 24x + 36 = 0\].

Identifying x-intercepts involves finding the roots or zeros of the equation. This is because when the value of the function, \(y\), is equal to zero, the line intersects the x-axis. On the calculator, this process involves using the "ZERO" function:

• First, graph the quadratic equation.
• Next, utilize the "ZERO" function to seek out these points of intersection.
• You must set bounds as a guided region for the calculator to identify where the curve crosses the x-axis accurately.

Once located, the x-intercepts give the solutions for the equation, rounded as directed. Thus, x-intercepts are crucial as they represent where solutions to the equation lie in their simplest form.

Quadratic Equations

Quadratic equations are polynomials of the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are coefficients. In our exercise, the equation \(4x^2 + 24x + 45 = 9\) needed adjusting to this form.

By subtracting 9 from both sides, the equation takes the standard quadratic form \(4x^2 + 24x + 36 = 0\). This is the foundation for further operations, such as graphing or solving:

• The quadratic equation usually creates a parabola when graphed.
• The shape is determined by the sign of \(a\): a positive \(a\) means the parabola opens upwards, while a negative \(a\) means it opens downwards.
• Solutions of the equation, like where it meets the x-axis, are found through its intercepts.

Understanding quadratic equations enables smoother analysis and solutions when visualizing them with a graphing calculator.

Graphing Techniques

To effectively use a graphing calculator for solving equations, some fundamental graphing techniques come into play. Setting up an appropriate window is the first step to capturing the entirety of the quadratic equation's curve.

• The suggested window setting, \([-10, 10]\) for both axes, ensures the graph fits within a visible range.
• If the graph isn’t visible, adjusting and zooming in using features such as "ZOOM IN" are necessary methods for clarifying intersections.
• Using a combination of "TRACE" and "SOLVE" functions helps pinpoint the exact values, as they bring precision to tracking the nature of the curve at particular points.

These graphing techniques simplify the exploration process, making tools like graphing calculators insightful companions for solving quadratics. Practicing these techniques makes interpreting graphs less challenging and facilitates finding solutions like x-intercepts more efficiently.