Problem 61 - 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 the Graphing Calculator Basics appendix, page A2) and use ZERO or TRACE and ZOOM IN.] Round answers to two decimal places. $$ 2 x^{2}+3 x-6=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are approximately \(-3.00\) and \(1.00\).

Step 1: Understand the Equation

The given equation is a quadratic equation: \(2x^2 + 3x - 6 = 0\). Our goal is to find the values of \(x\) that make this equation true (the solution or roots of the equation).

Step 2: Set up the Graphing Calculator

Enter the equation \(y = 2x^2 + 3x - 6\) into your graphing calculator. Begin with the window setting set to \([-10, 10]\) on the x-axis and \([-10, 10]\) on the y-axis.

Step 3: Graph the Equation

After entering the equation, plot the graph. You should see a parabola that might intersect the x-axis at one or more points.

Step 4: Find the Roots

Use the 'ZERO' feature of the calculator to find where the graph crosses the x-axis. Alternatively, you can also use the 'TRACE' feature to locate when \(y\) value approaches zero, or use 'ZOOM IN' to closely examine potential root values.

Step 5: Record the Solutions

After finding the x-intercepts where the parabola crosses the x-axis, note these values. You will typically get two values for \(x\) due to the nature of quadratic equations.

Step 6: Round Answers

Once the solutions are located, round each x-intercept to two decimal places for precision. These are the solutions to the equation \(2x^2 + 3x - 6 = 0\).

Key Concepts

Quadratic Equation

A quadratic equation is a fundamental form of polynomial equations, characterized by its highest exponent of two. It takes the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The significance of the quadratic equation lies in its ability to model a wide range of real-world situations, from projectile motion to areas and more.
Quadratic equations are intriguing because they often have up to two solutions, known as roots. These solutions represent points where the quadratic curve, known as a parabola, intersects the x-axis. Understanding how to solve quadratic equations is a crucial skill in both algebra and calculus, paving the way for advanced studies and applications.
In the case of our exercise, the equation is \(2x^2 + 3x - 6 = 0\). We aim to find the solutions to this by determining where the curve of this equation meets the x-axis.

Roots of Equation

The roots of a quadratic equation are the solution values for \(x\) that satisfy the equation \( ax^2 + bx + c = 0 \). These values represent the x-coordinates of points where the graph of the quadratic function touches or crosses the x-axis.
Every quadratic equation potentially has two roots, one root, or no real roots. This outcome depends on the discriminant, \( b^2 - 4ac \).

• If the discriminant is positive, there are two distinct real roots.
• If it's zero, there is exactly one real root (or a repeated root).
• If it's negative, there are no real roots, but rather two complex roots.

Identifying the roots is vital for understanding the behavior of the quadratic function and solving the equation. Roots are essential, especially when applying mathematical methods to physical scenarios where determining potential outcomes or positions is necessary.

Graphing Parabolas

Graphing parabolas is a visual method that assists in understanding the behavior of quadratic equations. A parabola is a symmetric curve created by plotting a quadratic equation on a graph. The general shape is a "U" or an inverted "U," depending on the value of \(a\) in \( ax^2 + bx + c \).
For positive \(a\), the parabola opens upward, while for negative \(a\), it opens downward. The vertex of the parabola is its highest or lowest point, determined using the formula \( x = -\frac{b}{2a} \).
Using a graphing calculator to plot a parabola helps in identifying:

• The vertex, which indicates the peak or trough.
• The axis of symmetry, a vertical line passing through the vertex.
• The direction and width of the parabola.

Graphing makes it simple to spot the intersections with the x-axis, which are the real roots or zeroes of the quadratic function.

Finding Zeroes

Finding the zeroes of a quadratic function involves determining the x-values where the function evaluates to zero, which correspond to the roots of the equation. These zeroes are the x-intercepts of the parabola on the graph.
Graphing calculators are excellent tools for locating these zeroes. By entering the quadratic equation, users can employ features like 'ZERO,' 'TRACE,' and 'ZOOM IN' to identify the points where the graph crosses the x-axis.
Here's how you can effectively use these features:

• 'ZERO' is a dedicated tool specifically designed to find x-intercepts, providing precise zeroes where \(y = 0\).
• 'TRACE' allows you to move along the graph to identify approximate x-intercepts when the y-value gets close to zero. It's more of a manual check.
• 'ZOOM IN' aids by enhancing graph visibility, making it easier to see and identify intersections.

Locating these zeroes on the graph and subsequently rounding them gives you the solutions to the quadratic equation.