Solving Quadratic Inequality 2x^2 + 2x - 4 > (1/4)(x^2 + 4) A Step-by-Step Guide

Introduction: Understanding Quadratic Inequalities

In the realm of mathematics, quadratic inequalities play a crucial role in understanding the behavior of quadratic functions and their relationships with the x-axis. These inequalities extend the concept of quadratic equations by introducing a comparison, such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤), between a quadratic expression and a constant or another expression. In this comprehensive discussion, we will delve into the process of solving the quadratic inequality 2x² + 2x - 4 > (1/4)(x² + 4), providing a step-by-step guide that not only solves the specific problem but also elucidates the underlying principles applicable to a broader range of quadratic inequalities.

Before embarking on the solution, it's essential to grasp the fundamental concepts. A quadratic inequality, in its general form, can be expressed as ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, or ax² + bx + c ≤ 0, where 'a', 'b', and 'c' are constants and 'x' is the variable. The solutions to these inequalities are typically intervals or unions of intervals on the number line, representing the values of 'x' for which the inequality holds true. The process of solving quadratic inequalities involves several key steps, including simplifying the inequality, finding the roots of the corresponding quadratic equation, and determining the intervals where the inequality is satisfied. This exploration will not only enhance your problem-solving skills but also deepen your understanding of quadratic functions and their applications in various mathematical and real-world contexts. By mastering these techniques, you'll be well-equipped to tackle a wide array of problems involving quadratic inequalities, fostering a strong foundation in algebraic concepts.

Step 1: Simplify the Inequality

Our initial objective in tackling the inequality 2x² + 2x - 4 > (1/4)(x² + 4) is to simplify the expression, making it easier to manipulate and solve. This involves eliminating fractions and rearranging the terms to bring the inequality into a standard quadratic form. To begin, we multiply both sides of the inequality by 4, a strategic move that clears the fraction and sets the stage for further simplification. This multiplication yields the expression 8x² + 8x - 16 > x² + 4. The rationale behind this step is to work with integer coefficients, which often simplifies the subsequent algebraic manipulations.

Following the removal of the fraction, our next task is to consolidate all terms on one side of the inequality, effectively setting the other side to zero. This is a crucial step in transforming the inequality into a standard quadratic form, which is essential for identifying the coefficients and applying appropriate solution techniques. By subtracting x² and 4 from both sides of the inequality, we achieve this consolidation. This operation results in the inequality 7x² + 8x - 20 > 0. This form is now a standard quadratic inequality, where the quadratic expression is compared to zero. The coefficients are clearly visible (a = 7, b = 8, c = -20), and we are now in a position to analyze the quadratic expression and determine the values of x that satisfy the inequality. This simplified form is the foundation upon which we will build our solution, allowing us to apply methods such as factoring, completing the square, or using the quadratic formula to find the critical points that define the intervals of the solution.

Step 2: Find the Roots of the Quadratic Equation

Having simplified the inequality to the standard quadratic form 7x² + 8x - 20 > 0, the next crucial step is to find the roots of the corresponding quadratic equation 7x² + 8x - 20 = 0. The roots of this equation, also known as the zeros or x-intercepts, are the points where the quadratic function intersects the x-axis. These roots play a vital role in determining the intervals where the inequality holds true. There are several methods to find these roots, including factoring, completing the square, and applying the quadratic formula. In this case, we will employ the quadratic formula, a versatile tool that can solve any quadratic equation, regardless of whether it can be easily factored.

The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. For our equation, 7x² + 8x - 20 = 0, we have a = 7, b = 8, and c = -20. Substituting these values into the quadratic formula, we get x = (-8 ± √(8² - 4 * 7 * -20)) / (2 * 7). Simplifying the expression under the square root, we have 8² - 4 * 7 * -20 = 64 + 560 = 624. Thus, the roots are x = (-8 ± √624) / 14. Further simplification involves finding the square root of 624, which is approximately 24.98. Therefore, the roots are approximately x = (-8 + 24.98) / 14 ≈ 1.21 and x = (-8 - 24.98) / 14 ≈ -2.36. These roots are the critical points that divide the number line into intervals, which we will analyze in the next step to determine the solution set for the inequality.

Step 3: Determine the Intervals Where the Inequality Holds True

With the roots of the quadratic equation 7x² + 8x - 20 = 0 found to be approximately x ≈ 1.21 and x ≈ -2.36, we now proceed to the crucial step of determining the intervals where the original inequality, 7x² + 8x - 20 > 0, holds true. These roots act as critical points, dividing the number line into three distinct intervals: (-∞, -2.36), (-2.36, 1.21), and (1.21, ∞). To ascertain whether the inequality is satisfied in each interval, we employ a test point method. This involves selecting a representative value within each interval and substituting it into the inequality. The sign of the resulting expression will indicate whether the inequality is satisfied in that interval.

For the interval (-∞, -2.36), let's choose a test point x = -3. Substituting this into the inequality, we get 7(-3)² + 8(-3) - 20 = 63 - 24 - 20 = 19, which is greater than 0. Thus, the inequality holds true in this interval. For the interval (-2.36, 1.21), we select x = 0 as our test point. Substituting this, we get 7(0)² + 8(0) - 20 = -20, which is less than 0. Hence, the inequality is not satisfied in this interval. Finally, for the interval (1.21, ∞), we choose x = 2. Substituting this, we get 7(2)² + 8(2) - 20 = 28 + 16 - 20 = 24, which is greater than 0. Therefore, the inequality holds true in this interval. Based on this analysis, the solution to the inequality 7x² + 8x - 20 > 0 is the union of the intervals (-∞, -2.36) and (1.21, ∞). This means that all values of x less than -2.36 and all values of x greater than 1.21 satisfy the original inequality. The graphical interpretation of this solution involves the parabola representing the quadratic function lying above the x-axis in these intervals.

Step 4: Express the Solution

Having identified the intervals where the inequality 7x² + 8x - 20 > 0 holds true, our final step is to express the solution in a clear and concise manner. As determined in the previous step, the solution set consists of the union of two intervals: (-∞, -2.36) and (1.21, ∞). This means that any value of x within these intervals will satisfy the original inequality, 2x² + 2x - 4 > (1/4)(x² + 4). To express this solution formally, we use interval notation, which provides a compact and unambiguous way to represent sets of numbers.

The solution can be written as x ∈ (-∞, -2.36) ∪ (1.21, ∞). The symbol '∈' indicates that x belongs to the specified set, and the symbol '∪' represents the union of two sets. The parentheses '(' and ')' indicate that the endpoints -2.36 and 1.21 are not included in the solution set, as the inequality is strict ('>' rather than '≥'). The symbols '-∞' and '∞' represent negative infinity and positive infinity, respectively, indicating that the intervals extend indefinitely in the negative and positive directions. This notation succinctly captures all values of x that satisfy the inequality. It's important to note that the accuracy of the endpoints depends on the precision of the roots calculated in Step 2. In practical applications, the solution may be rounded to a suitable number of decimal places. This final representation provides a complete and precise answer to the problem, showcasing the set of all possible solutions in a standardized mathematical notation.

Conclusion: Mastering Quadratic Inequalities

In this detailed exploration, we have successfully navigated the process of solving the quadratic inequality 2x² + 2x - 4 > (1/4)(x² + 4). By methodically simplifying the inequality, finding the roots of the corresponding quadratic equation, determining the intervals where the inequality holds true, and expressing the solution in interval notation, we have not only solved the specific problem but also elucidated the general principles applicable to solving quadratic inequalities. This comprehensive approach underscores the importance of a systematic methodology in mathematical problem-solving, emphasizing the need to break down complex problems into manageable steps.

The key takeaways from this discussion include the significance of simplifying the inequality to a standard quadratic form, the crucial role of the quadratic formula in finding the roots of the equation, and the effectiveness of the test point method in determining the intervals that satisfy the inequality. Furthermore, the use of interval notation provides a concise and standardized way to express the solution set. Mastering these techniques is essential for students and professionals alike, as quadratic inequalities appear in various mathematical and real-world contexts. From optimizing engineering designs to modeling economic trends, the ability to solve quadratic inequalities is a valuable skill. By understanding the underlying principles and practicing problem-solving, one can develop a strong foundation in quadratic inequalities and confidently tackle a wide range of related problems. This mastery not only enhances mathematical proficiency but also fosters critical thinking and analytical skills that are applicable across diverse fields.