Solving Quadratic Inequality Find Range Of N For 3 + 14n - 5n² ≤ 0
In the realm of mathematics, inequalities play a crucial role in defining the boundaries and constraints of solutions. This article delves into the intricacies of solving a quadratic inequality, specifically focusing on finding the range of values for 'n' that satisfy the inequality 3 + 14n - 5n² ≤ 0. This exploration will not only provide a step-by-step solution but also shed light on the underlying concepts and techniques involved in handling such mathematical expressions. Understanding these concepts is fundamental for students, educators, and anyone with a passion for mathematical problem-solving. We'll start by rearranging the inequality into a standard quadratic form, making it easier to analyze its properties. Then, we'll discuss the importance of finding the roots of the corresponding quadratic equation, as these roots act as critical points that divide the number line into intervals. The sign of the quadratic expression within each interval will determine whether the inequality is satisfied. Finally, we'll present the solution set, clearly indicating the range of 'n' values that make the inequality true. This journey through the solution process will not only enhance your algebraic skills but also deepen your understanding of quadratic functions and their graphical representations. By the end of this article, you will be equipped with the knowledge and confidence to tackle similar inequality problems with ease.
Transforming the Inequality
The first step in solving any inequality is to bring it into a standard form that makes it easier to analyze. For a quadratic inequality like 3 + 14n - 5n² ≤ 0, the standard form is ax² + bx + c ≤ 0 (or ≥ 0, < 0, > 0), where a, b, and c are constants. In our case, we need to rearrange the given inequality to match this standard form. This involves rearranging the terms so that the term with the highest power of 'n' comes first, followed by the term with 'n', and finally the constant term. This seemingly simple rearrangement is crucial because it allows us to easily identify the coefficients a, b, and c, which are essential for further analysis. Specifically, 'a' is the coefficient of the n² term, 'b' is the coefficient of the 'n' term, and 'c' is the constant term. Once we have these coefficients, we can apply various techniques, such as factoring or using the quadratic formula, to find the roots of the corresponding quadratic equation. These roots play a pivotal role in determining the intervals where the inequality holds true. Moreover, the standard form provides a clear visual representation of the quadratic function, allowing us to sketch its graph and understand its behavior. This visual understanding can be incredibly helpful in solving the inequality, as we can see where the function is positive, negative, or zero. So, let's begin by rearranging the terms of the inequality to bring it into the standard quadratic form, setting the stage for the next steps in our solution.
Rearranging the terms, we get:
-5n² + 14n + 3 ≤ 0
To simplify the process, we can multiply the entire inequality by -1. Remember, when multiplying or dividing an inequality by a negative number, we must reverse the inequality sign. This gives us:
5n² - 14n - 3 ≥ 0
Now, we have a quadratic inequality in the standard form, making it easier to proceed with finding the solution.
Finding the Roots: The Key to Unlocking the Solution
Now that we have our quadratic inequality in standard form (5n² - 14n - 3 ≥ 0), the next crucial step is to find the roots of the corresponding quadratic equation. These roots are the values of 'n' that make the equation 5n² - 14n - 3 = 0 true. They are the points where the graph of the quadratic function intersects the n-axis, and they act as critical dividing points on the number line. The intervals created by these roots are where the quadratic expression will either be positive, negative, or zero. To find these roots, we have a couple of options: factoring or using the quadratic formula. Factoring involves expressing the quadratic expression as a product of two linear factors. This method is efficient when the quadratic expression can be easily factored. However, when factoring proves difficult, the quadratic formula provides a reliable alternative. The quadratic formula is a general solution that works for any quadratic equation, regardless of whether it can be factored easily. It states that for an equation of the form ax² + bx + c = 0, the roots are given by: n = (-b ± √(b² - 4ac)) / 2a. In our case, a = 5, b = -14, and c = -3. Plugging these values into the quadratic formula will give us the roots of our equation. These roots are the cornerstone of our solution, as they define the intervals we need to consider to determine the range of 'n' values that satisfy the original inequality. So, let's delve into the process of finding these roots, either by factoring or by applying the quadratic formula, and unlock the key to solving our inequality.
We can use the quadratic formula to find the roots of the equation 5n² - 14n - 3 = 0. The quadratic formula is given by:
n = (-b ± √(b² - 4ac)) / 2a
Where a = 5, b = -14, and c = -3. Substituting these values into the formula, we get:
n = (14 ± √((-14)² - 4 * 5 * -3)) / (2 * 5) n = (14 ± √(196 + 60)) / 10 n = (14 ± √256) / 10 n = (14 ± 16) / 10
This gives us two roots:
n₁ = (14 + 16) / 10 = 3 n₂ = (14 - 16) / 10 = -1/5
Therefore, the roots of the equation are n = 3 and n = -1/5. These roots are crucial as they divide the number line into intervals that we will analyze further.
Analyzing Intervals: Determining Where the Inequality Holds True
With the roots of the quadratic equation in hand (n = 3 and n = -1/5), we now have the critical points that divide the number line into distinct intervals. These intervals are the key to determining the range of 'n' values that satisfy our original inequality, 5n² - 14n - 3 ≥ 0. The roots act as boundaries, separating the number line into regions where the quadratic expression will either be positive, negative, or zero. To analyze these intervals, we need to test a value within each interval in the quadratic expression. By substituting a test value from each interval into the expression 5n² - 14n - 3, we can determine the sign of the expression in that interval. If the result is positive, the inequality is satisfied in that interval. If the result is negative, the inequality is not satisfied. This process of testing intervals allows us to systematically map out the regions where the inequality holds true. For example, we might choose a test value less than -1/5, a value between -1/5 and 3, and a value greater than 3. By evaluating the quadratic expression at these test points, we can build a clear picture of its behavior across the entire number line. This interval analysis is a fundamental technique in solving inequalities, providing a structured approach to identifying the solution set. It transforms the problem from a symbolic expression into a concrete analysis of regions on the number line, making the solution more intuitive and accessible. So, let's embark on this process of analyzing the intervals, testing values, and uncovering the range of 'n' that satisfies our inequality.
These roots divide the number line into three intervals: (-∞, -1/5], [-1/5, 3], and [3, ∞). We will now test a value from each interval to determine the sign of the quadratic expression 5n² - 14n - 3 in that interval.
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Interval (-∞, -1/5]: Let's test n = -1: 5(-1)² - 14(-1) - 3 = 5 + 14 - 3 = 16 > 0 The inequality holds true in this interval.
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Interval [-1/5, 3]: Let's test n = 0: 5(0)² - 14(0) - 3 = -3 < 0 The inequality does not hold true in this interval.
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Interval [3, ∞): Let's test n = 4: 5(4)² - 14(4) - 3 = 80 - 56 - 3 = 21 > 0 The inequality holds true in this interval.
Solution Set: Defining the Range of n
After meticulously analyzing the intervals created by the roots of our quadratic equation, we have arrived at the final stage of our journey: defining the solution set. The solution set represents the range of 'n' values that satisfy the original inequality, 5n² - 14n - 3 ≥ 0. Our interval analysis revealed that the inequality holds true in two distinct regions: the interval from negative infinity up to and including -1/5, and the interval from 3 up to positive infinity. These intervals represent the values of 'n' for which the quadratic expression is either positive or zero, fulfilling the condition of our inequality. To express this solution set mathematically, we use interval notation, a concise way of representing a range of values. Interval notation uses brackets and parentheses to indicate whether the endpoints of an interval are included or excluded. Square brackets ([ and ]) indicate that the endpoint is included in the interval, while parentheses (( and )) indicate that the endpoint is excluded. Infinity (∞) and negative infinity (-∞) are always enclosed in parentheses, as they are not specific numbers but rather concepts representing unboundedness. Therefore, the solution set for our inequality is the union of the two intervals where it holds true. This union combines the two intervals into a single set, representing all the 'n' values that satisfy the inequality. The solution set is the definitive answer to our problem, providing a clear and precise description of the range of 'n' values that make the inequality true. So, let's formally express the solution set using interval notation, completing our exploration of this quadratic inequality.
Based on the interval analysis, the solution set for the inequality 5n² - 14n - 3 ≥ 0 is:
n ∈ (-∞, -1/5] ∪ [3, ∞)
This means that the inequality is satisfied when 'n' is less than or equal to -1/5 or greater than or equal to 3. This completes our solution for finding the range of values of 'n' that satisfy the given inequality.
Conclusion: Mastering Quadratic Inequalities
In this comprehensive exploration, we embarked on a journey to solve the quadratic inequality 3 + 14n - 5n² ≤ 0. We navigated through the essential steps, from transforming the inequality into standard form to meticulously analyzing intervals and ultimately defining the solution set. We began by recognizing the importance of rearranging the inequality into the standard quadratic form, which allowed us to easily identify the coefficients and apply appropriate solution techniques. We then delved into the crucial process of finding the roots of the corresponding quadratic equation, understanding that these roots act as critical dividing points on the number line. The quadratic formula proved to be an invaluable tool in this step, providing a reliable method for finding the roots regardless of whether the expression could be easily factored. With the roots in hand, we proceeded to analyze the intervals they created on the number line, testing values within each interval to determine the sign of the quadratic expression. This systematic approach allowed us to map out the regions where the inequality held true. Finally, we synthesized our findings and expressed the solution set using interval notation, a concise and precise way of representing the range of 'n' values that satisfy the inequality. The solution set, n ∈ (-∞, -1/5] ∪ [3, ∞), represents the culmination of our efforts, providing a definitive answer to the problem. This journey through solving a quadratic inequality has not only enhanced our algebraic skills but also deepened our understanding of quadratic functions and their behavior. The techniques and concepts explored in this article are applicable to a wide range of mathematical problems, empowering us to tackle future challenges with confidence and proficiency. Mastering quadratic inequalities is a fundamental step in mathematical problem-solving, and the knowledge gained here will serve as a strong foundation for further exploration in the world of mathematics.
