Solving Quadratic Inequalities Factored Form Explained

Finding the right inequality in factored form can be tricky, especially when dealing with quadratic functions. In this comprehensive guide, we'll break down the process step by step, ensuring you understand how to represent a region greater than or equal to a quadratic function. We'll tackle a specific problem involving zeros and a boundary point, providing clear explanations and strategies along the way. This article aims to help you master this essential mathematical concept.

Understanding Quadratic Functions

Before diving into the inequality, it's crucial to grasp the basics of quadratic functions. A quadratic function is generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0).

Key Characteristics of Quadratic Functions

To effectively work with quadratic functions, understanding their key characteristics is essential:

• Zeros (Roots or x-intercepts): These are the points where the parabola intersects the x-axis. At these points, f(x) = 0. Zeros are crucial in determining the factored form of a quadratic function.
• Vertex: This is the highest or lowest point on the parabola, depending on whether the parabola opens downwards or upwards, respectively. The vertex represents the maximum or minimum value of the function.
• Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation of the axis of symmetry is x = h, where (h, k) is the vertex.
• Y-intercept: This is the point where the parabola intersects the y-axis. It's found by setting x = 0 in the quadratic function.

Factored Form of a Quadratic Function

The factored form of a quadratic function is expressed as f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the zeros of the function, and a is a constant that determines the parabola's direction and width. This form is particularly useful when the zeros are known, as it directly incorporates this information.

When you're dealing with quadratic inequalities, the factored form becomes even more significant. It helps in visualizing and determining the regions where the inequality holds true. For instance, if you need to find the region where f(x) ≥ 0, the factored form assists in identifying intervals where the function's values are non-negative.

Consider a scenario where you have a quadratic function with zeros -3.5 and 11.5. The factored form immediately gives you a structure to work with: f(x) = a(x + 3.5)(x - 11.5). The remaining task is to find the value of a and determine the inequality sign. This is where additional information, such as a point on the boundary, becomes crucial.

Understanding the factored form not only simplifies the representation of quadratic functions but also lays the groundwork for solving inequalities and understanding the graphical behavior of parabolas. The ability to seamlessly transition between standard form, vertex form, and factored form is a hallmark of strong quadratic function comprehension.

Problem Statement and Setup

Let's consider the specific problem we aim to solve: Which inequality in factored form represents the region greater than or equal to the quadratic function with zeros -3.5 and 11.5 and includes the point (8.5, -54) on the boundary? This problem combines several key concepts, including quadratic functions, factored form, inequalities, and boundary points.

Breaking Down the Problem

To tackle this problem effectively, we need to break it down into smaller, manageable steps. This approach ensures clarity and reduces the chances of errors. Here’s a step-by-step breakdown:

1. Identify the Zeros: The zeros of the quadratic function are given as -3.5 and 11.5. These are the points where the parabola intersects the x-axis.
2. Write the Factored Form: Using the zeros, we can express the quadratic function in factored form. The general form is y = a(x - r₁)(x - r₂), where r₁ and r₂ are the zeros. Plugging in the given zeros, we get y = a(x + 3.5)(x - 11.5).
3. Determine the Constant a: We are given a point (8.5, -54) on the boundary. This means that when x = 8.5, y = -54. We can use this information to solve for the constant a.
4. Write the Inequality: The problem asks for the region greater than or equal to the quadratic function. This means we need to determine the correct inequality sign (≥) and incorporate it into our equation.

Setting Up the Equation

Using the factored form and the given point, we can set up an equation to solve for a. Substituting x = 8.5 and y = -54 into the equation y = a(x + 3.5)(x - 11.5), we get:

-54 = a(8.5 + 3.5)(8.5 - 11.5)

This equation will allow us to find the value of a, which is crucial for determining the specific quadratic function. Understanding how to set up this equation is a fundamental step in solving problems involving quadratic functions and inequalities.

Once we find the value of a, we can write the complete factored form of the quadratic function. The next step involves translating this function into an inequality that represents the desired region. This requires careful consideration of the direction of the inequality (greater than or equal to) and the overall shape of the parabola.

By meticulously setting up the equation and understanding the problem’s requirements, we lay a solid foundation for solving the inequality. This structured approach is invaluable in mathematics, where clarity and precision are paramount.

Solving for the Constant a

Now, let’s proceed with solving for the constant a. We have the equation:

-54 = a(8.5 + 3.5)(8.5 - 11.5)

Step-by-Step Solution

To find a, we need to simplify and solve this equation. Here's a detailed step-by-step solution:

1. Simplify the Parentheses: First, simplify the expressions inside the parentheses:

   • 8.5 + 3.5 = 12
   • 8.5 - 11.5 = -3

   So, the equation becomes: -54 = a(12)(-3)

2. Multiply the Constants: Next, multiply the constants on the right side of the equation:

   • (12)(-3) = -36

   The equation is now: -54 = -36a

3. Solve for a: To isolate a, divide both sides of the equation by -36:

   • a = -54 / -36
   • a = 3/2

   Therefore, the value of a is 3/2.

Significance of the Constant a

The value of a is crucial because it determines the shape and direction of the parabola. In this case, a = 3/2, which is positive. This tells us that the parabola opens upwards. The magnitude of a also affects the parabola's width; a larger absolute value of a results in a narrower parabola, while a smaller value results in a wider one.

Plugging a Back into the Factored Form

Now that we have found a, we can plug it back into the factored form of the quadratic function. The factored form is y = a(x + 3.5)(x - 11.5). Substituting a = 3/2, we get:

y = (3/2)(x + 3.5)(x - 11.5)

This is the specific quadratic function in factored form that has zeros at -3.5 and 11.5 and passes through the point (8.5, -54). Understanding how to solve for a and incorporate it into the factored form is a critical skill in dealing with quadratic functions and inequalities. This step sets the stage for writing the inequality that represents the desired region.

Writing the Inequality

Now that we have the factored form of the quadratic function, y = (3/2)(x + 3.5)(x - 11.5), the next step is to write the inequality that represents the region greater than or equal to this function. The problem specifies that we are looking for the region greater than or equal to the function, which means we need to use the ≥ sign.

Understanding the Inequality Sign

The inequality y ≥ (3/2)(x + 3.5)(x - 11.5) means that we are interested in all the points (x, y) that lie on or above the parabola defined by the quadratic function. The “equal to” part of the inequality includes the points on the parabola itself, which is the boundary of the region.

Formulating the Inequality

To write the inequality, we simply replace the equals sign in the equation with the greater than or equal to sign:

y ≥ (3/2)(x + 3.5)(x - 11.5)

This inequality represents the region that includes all points on or above the parabola defined by the quadratic function y = (3/2)(x + 3.5)(x - 11.5). It’s important to understand that the parabola itself is part of the solution set because of the “or equal to” condition.

Visualizing the Region

Visualizing this inequality can be helpful. Imagine the parabola opening upwards, with zeros at -3.5 and 11.5. The region represented by the inequality y ≥ (3/2)(x + 3.5)(x - 11.5) is the area above the parabola, including the parabola itself. This region extends infinitely upwards and to the sides.

Importance of the Inequality

The inequality y ≥ (3/2)(x + 3.5)(x - 11.5) is the final answer to the problem. It provides a concise mathematical representation of the region that satisfies the given conditions. This process demonstrates how to translate a description of a region in relation to a quadratic function into an algebraic inequality, a fundamental skill in mathematics.

In summary, writing the inequality involves understanding the relationship between the function and the region, choosing the correct inequality sign, and expressing the region algebraically. This inequality not only answers the specific problem but also provides a powerful tool for representing and analyzing regions defined by quadratic functions.

Final Answer and Conclusion

The Solution

After working through the steps, we have arrived at the final answer. The inequality in factored form that represents the region greater than or equal to the quadratic function with zeros -3.5 and 11.5 and includes the point (8.5, -54) on the boundary is:

y ≥ (3/2)(x + 3.5)(x - 11.5)

This inequality succinctly captures all the given conditions. It represents a region that includes the parabola itself and all points above it. The factored form clearly shows the zeros of the quadratic function, and the constant a = 3/2 determines the shape and direction of the parabola.

Recap of the Steps

To recap, we followed these steps to solve the problem:

1. Identified the Zeros: We recognized that the zeros were given as -3.5 and 11.5.
2. Wrote the Factored Form: We expressed the quadratic function in factored form using the zeros: y = a(x + 3.5)(x - 11.5).
3. Solved for the Constant a: We used the given point (8.5, -54) to solve for a, finding that a = 3/2.
4. Wrote the Inequality: We formulated the inequality y ≥ (3/2)(x + 3.5)(x - 11.5), representing the region greater than or equal to the quadratic function.

Importance of Understanding Quadratic Inequalities

Understanding quadratic inequalities is crucial in various fields, including mathematics, physics, and engineering. They are used to model and solve problems involving optimization, constraints, and regions defined by quadratic functions. The ability to translate a problem description into an algebraic inequality is a valuable skill.

Concluding Thoughts

In conclusion, solving quadratic inequalities in factored form involves a systematic approach, from identifying key information to formulating the inequality. This article has provided a detailed guide on how to tackle such problems, emphasizing the importance of understanding the properties of quadratic functions and inequalities. With practice, you can master these concepts and confidently solve a wide range of mathematical problems.

This structured approach, combining algebraic manipulation with conceptual understanding, empowers you to tackle complex problems involving quadratic functions and inequalities effectively. The final inequality not only solves the given problem but also provides a powerful tool for analyzing and representing regions in mathematical contexts.