Problem 23 - Algebra 2

Question

Solve each equation by graphing. Give each answer to at most two decimal places. $$ 6 x^{2}=-19 x-15 $$

Step-by-Step Solution

Answer

The solutions are the x-intercepts of the graph of the equation, rounded to two decimal places.

Step 1: Rewrite the equation in standard form

To get the equation into standard quadratic form, we need to move all terms to one side of the equation. We achieve this by adding 19x and 15 to both sides to get: $$0 = 6x^2 + 19x + 15$$

Step 2: Graph the quadratic function

Using graphing technology or software, graph the function $$f(x) = 6x^2 + 19x + 15$$Make sure to plot the function over an interval that includes the x-intercepts, if they exist.

Step 3: Identify the x-intercepts

The x-intercepts are where the function crosses the x-axis. These points correspond to the solutions of the equation. Look for the x-values where the graph intersects the x-axis and record them.

Step 4: Approximate the solutions

If necessary, adjust the view or use the 'trace' or 'zero' function to get a more accurate estimate of the solutions. Round your answers to at most two decimal places.

Key Concepts

Quadratic Functions

A quadratic function represents a parabola on a coordinate plane and is typically expressed in the general form f(x) = ax^2 + bx + c, where a, b, and c are real numbers with a eq 0. Its graph opens upward if a is positive, and downward if a is negative. Quadratic functions are fundamental in mathematics as they model various natural and physical phenomena. They are characterized by specific features such as a vertex, axis of symmetry, and intercepts, which can be found using algebraic methods or graphing technology.

Understanding the behavior of quadratic functions is essential for solving quadratic equations, predicting outcomes in real-world scenarios, and analyzing the motion of objects under gravity, among other applications.

X-Intercepts

The x-intercepts of a quadratic function are the points where the graph of the function intersects the x-axis. In other words, they represent the values of x for which f(x) = 0. These points are significant, as they are also known as the 'roots' or 'solutions' of the quadratic equation. A quadratic function can have zero, one, or two real x-intercepts, which is determined by the discriminant b^2 - 4ac. If the discriminant is positive, there are two distinct intercepts; if it's zero, there's exactly one intercept (also known as the vertex); and if negative, there are no real x-intercepts, indicating the parabola does not cross the x-axis.

These intercepts can be found by factoring, completing the square, using the quadratic formula, or through graphing. When solving by graphing, precision in identifying these points is key, as it gives accurate solutions to the equation.

Standard Quadratic Form

The standard quadratic form of an equation is ax^2 + bx + c = 0, where a, b, and c are coefficients. To solve a quadratic equation, it is often helpful to first rewrite it in this format. This not only standardizes the approach for solution methods but also simplifies the use of tools such as the quadratic formula, x = (-b \[ \pm \] \sqrt{b^2 - 4ac})/(2a), which requires coefficients to be identified clearly. Additionally, when graphing, having the equation in standard form allows for easier manipulation and faster comprehension of the curve's characteristics, including its vertex and orientation.

Graphing Technology

Graphing technology, which can include graphing calculators or computer software, provides interactive ways to visualize and solve mathematical problems, particularly for quadratic functions. Tools such as Desmos, GeoGebra, or graphing calculators allow users to plot the parabolic graph of these functions with precision.

Utilizing these technologies simplifies the process of identifying key features like x-intercepts, vertices, and general shape. Moreover, these platforms typically offer functionalities like zooming, tracing points, or finding exact coordinates that enhance our ability to approximate solutions accurately. They are an invaluable resource in educational settings, helping students and teachers alike to better grasp the intricacies of quadratic equations and their solutions.

Approximating Solutions

Using Graphing Technology

The nature of graphing technology means that sometimes, we cannot get exact solutions from the graph, especially when dealing with irrational numbers. Instead, we approximate the solutions by identifying the points where the graph intersects the x-axis. These approximations are usually to a certain number of decimal places, as required by the context of the problem.

Importance of Precision

When approximating solutions, it's critical to ensure the highest possible precision within the constraints of the available technology. This can be done by altering the graph's viewing window for a closer look or by using built-in features to pinpoint the intercepts. Accurate approximations are crucial in fields like engineering and physics, where even minor deviations can lead to significant consequences.