Finding K For Equal Roots In Quadratic Equation X^2 + 4kx + (k^2 - K + 2) = 0

In the realm of mathematics, particularly in algebra, quadratic equations hold a significant position. These equations, characterized by the presence of a squared term, often present intriguing challenges and solutions. One such challenge involves determining the conditions under which a quadratic equation possesses equal roots. This exploration delves into the intricacies of finding the value of 'k' that satisfies the condition of equal roots in the quadratic equation: x^2 + 4kx + (k^2 - k + 2) = 0. Understanding the nature of roots in quadratic equations is crucial for various applications in physics, engineering, and computer science. The discriminant, a key component of the quadratic formula, plays a pivotal role in determining the nature of roots. When the discriminant is zero, the quadratic equation has equal real roots. This article will systematically guide you through the process of calculating the discriminant and setting it to zero to solve for 'k', thereby providing a comprehensive understanding of how to approach similar problems. We will also discuss the underlying principles that govern quadratic equations and their roots, ensuring a solid foundation for further mathematical explorations. This problem not only tests your algebraic skills but also enhances your analytical thinking and problem-solving abilities.

Before diving into the specific problem, let's establish a firm understanding of quadratic equations and their roots. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The roots of a quadratic equation are the values of 'x' that satisfy the equation. These roots represent the points where the parabola described by the quadratic equation intersects the x-axis. Quadratic equations can have two distinct real roots, one repeated real root (equal roots), or two complex roots. The nature of the roots is determined by the discriminant, denoted as Δ (Delta), which is given by the formula Δ = b^2 - 4ac. The discriminant provides valuable information about the roots: If Δ > 0, the equation has two distinct real roots. If Δ = 0, the equation has one repeated real root (equal roots). If Δ < 0, the equation has two complex roots. In our case, we are interested in the condition where the roots are equal, which means we need to find the value of 'k' that makes the discriminant equal to zero. Understanding the relationship between the discriminant and the roots is fundamental to solving this problem. The coefficients 'a', 'b', and 'c' in the quadratic equation play a crucial role in determining the nature and values of the roots. By manipulating these coefficients, we can control the shape and position of the parabola, and consequently, the roots of the equation. This concept is widely used in various fields, such as designing parabolic reflectors in telescopes or optimizing trajectories in physics. Therefore, a thorough understanding of quadratic equations and their roots is essential for both academic and practical applications.

Our primary objective is to determine the value(s) of 'k' for which the given quadratic equation, x^2 + 4kx + (k^2 - k + 2) = 0, has equal roots. As we established earlier, a quadratic equation has equal roots when its discriminant (Δ) is equal to zero. In this specific equation, we can identify the coefficients as follows: a = 1, b = 4k, and c = (k^2 - k + 2). These coefficients are crucial for calculating the discriminant, which will then be set to zero to find the desired values of 'k'. The problem requires us to apply our knowledge of quadratic equations and the discriminant to solve for an unknown parameter. This type of problem is common in algebra and serves as a good exercise in applying theoretical concepts to practical scenarios. The challenge lies in correctly identifying the coefficients and performing the algebraic manipulations to solve the resulting equation. By solving this problem, we not only reinforce our understanding of quadratic equations but also develop our problem-solving skills. The ability to identify patterns, apply formulas, and manipulate equations is essential for success in mathematics and related fields. Furthermore, this problem highlights the importance of paying attention to detail and avoiding common algebraic errors. A small mistake in identifying the coefficients or simplifying the equation can lead to an incorrect solution. Therefore, it is crucial to approach the problem systematically and double-check each step to ensure accuracy. The process of finding 'k' involves setting up an equation based on the discriminant and then solving for 'k'. This process often requires algebraic manipulation, such as expanding squares, combining like terms, and factoring. These skills are fundamental to solving a wide range of mathematical problems, making this problem an excellent learning opportunity.

To find the value of 'k' for which the given quadratic equation has equal roots, our first step is to calculate the discriminant (Δ). Recall that the discriminant is given by the formula Δ = b^2 - 4ac. In the equation x^2 + 4kx + (k^2 - k + 2) = 0, we have identified the coefficients as: a = 1, b = 4k, and c = (k^2 - k + 2). Now, we substitute these values into the discriminant formula: Δ = (4k)^2 - 4 * 1 * (k^2 - k + 2) Next, we simplify the expression: Δ = 16k^2 - 4(k^2 - k + 2) Δ = 16k^2 - 4k^2 + 4k - 8 Δ = 12k^2 + 4k - 8 This simplified expression for the discriminant is a quadratic equation in terms of 'k'. To find the values of 'k' that result in equal roots for the original quadratic equation, we need to set this discriminant equal to zero and solve for 'k'. The correct calculation of the discriminant is crucial, as any error in this step will propagate through the rest of the solution. It's important to pay close attention to the signs and ensure that all terms are correctly multiplied and combined. The process of calculating the discriminant involves basic algebraic operations such as squaring, multiplication, and subtraction. These operations are fundamental to algebra, and proficiency in them is essential for solving more complex problems. The calculated discriminant, 12k^2 + 4k - 8, is a quadratic expression itself, which means we will need to use techniques for solving quadratic equations to find the values of 'k'. This highlights the interconnectedness of mathematical concepts and the importance of mastering foundational skills. The discriminant serves as a bridge between the coefficients of a quadratic equation and the nature of its roots, making it a powerful tool for analyzing and solving quadratic equations.

Now that we have calculated the discriminant as Δ = 12k^2 + 4k - 8, our next step is to set it equal to zero and solve for 'k'. This is because the condition for equal roots in a quadratic equation is that the discriminant must be zero. So, we have the equation: 12k^2 + 4k - 8 = 0 To simplify this equation, we can divide all terms by the greatest common divisor, which is 4: 3k^2 + k - 2 = 0 This simplified quadratic equation can be solved using various methods, such as factoring, completing the square, or the quadratic formula. In this case, factoring is a straightforward approach. We are looking for two numbers that multiply to (3 * -2 = -6) and add up to 1. These numbers are 3 and -2. So, we can rewrite the middle term as: 3k^2 + 3k - 2k - 2 = 0 Now, we factor by grouping: 3k(k + 1) - 2(k + 1) = 0 (3k - 2)(k + 1) = 0 Setting each factor equal to zero gives us the solutions for 'k': 3k - 2 = 0 => k = 2/3 k + 1 = 0 => k = -1 Therefore, the values of 'k' for which the given quadratic equation has equal roots are k = 2/3 and k = -1. This solution demonstrates the power of algebraic manipulation and factoring in solving quadratic equations. The ability to factor quadratic expressions is a fundamental skill in algebra and is essential for solving a wide range of problems. The process of setting the discriminant to zero and solving for the unknown parameter is a common technique in mathematics and is used in various contexts, such as optimization problems and curve fitting. The two values of 'k' we found indicate that there are two different quadratic equations, corresponding to these values of 'k', that have equal roots. This highlights the fact that quadratic equations can have different solutions depending on the parameters involved. The solutions k = 2/3 and k = -1 provide specific conditions under which the original quadratic equation will have equal roots, making the problem a valuable exercise in understanding the relationship between coefficients, discriminants, and roots of quadratic equations.

To ensure the correctness of our solutions, it's crucial to verify that the values of k = 2/3 and k = -1 indeed result in equal roots for the given quadratic equation x^2 + 4kx + (k^2 - k + 2) = 0. Let's start by substituting k = 2/3 into the equation: x^2 + 4(2/3)x + ((2/3)^2 - (2/3) + 2) = 0 x^2 + (8/3)x + (4/9 - 2/3 + 2) = 0 To simplify, we find a common denominator for the constant term: x^2 + (8/3)x + (4/9 - 6/9 + 18/9) = 0 x^2 + (8/3)x + (16/9) = 0 To eliminate fractions, we multiply the entire equation by 9: 9x^2 + 24x + 16 = 0 This quadratic equation can be factored as: (3x + 4)^2 = 0 This gives us a repeated root of x = -4/3, confirming that k = 2/3 results in equal roots. Now, let's substitute k = -1 into the equation: x^2 + 4(-1)x + ((-1)^2 - (-1) + 2) = 0 x^2 - 4x + (1 + 1 + 2) = 0 x^2 - 4x + 4 = 0 This quadratic equation can be factored as: (x - 2)^2 = 0 This gives us a repeated root of x = 2, confirming that k = -1 also results in equal roots. The verification process is an essential step in problem-solving, as it helps to identify any potential errors and ensures the accuracy of the solutions. By substituting the values of 'k' back into the original equation and showing that they lead to repeated roots, we have strengthened our confidence in the correctness of our solution. This process also reinforces our understanding of the relationship between the value of 'k', the coefficients of the quadratic equation, and the nature of its roots. The ability to verify solutions is a valuable skill in mathematics and other fields, as it promotes critical thinking and attention to detail. It also helps to develop a deeper understanding of the concepts involved and builds confidence in one's problem-solving abilities. Furthermore, verification can sometimes reveal alternative approaches or insights into the problem, leading to a more comprehensive understanding of the underlying principles.

In conclusion, we successfully determined the values of 'k' for which the quadratic equation x^2 + 4kx + (k^2 - k + 2) = 0 has equal roots. By applying the concept of the discriminant, we calculated Δ = 12k^2 + 4k - 8 and set it to zero to find the values of 'k'. Solving the resulting quadratic equation, we obtained k = 2/3 and k = -1 as the solutions. We then verified these solutions by substituting them back into the original equation and confirming that they indeed lead to equal roots. This problem provided a comprehensive exercise in applying the properties of quadratic equations and the discriminant. It reinforced our understanding of how the discriminant determines the nature of the roots and how to manipulate algebraic expressions to solve for unknown parameters. The ability to solve such problems is crucial in various fields, including mathematics, physics, engineering, and computer science. The systematic approach we used, involving calculating the discriminant, setting it to zero, solving for 'k', and verifying the solutions, is a valuable problem-solving strategy that can be applied to a wide range of mathematical problems. This problem also highlighted the importance of accuracy and attention to detail in algebraic manipulations. A small error in calculating the discriminant or solving the equation can lead to incorrect solutions. Therefore, it is essential to approach problems systematically and double-check each step to ensure accuracy. Furthermore, the verification process emphasized the importance of critical thinking and the need to confirm the validity of solutions. By verifying our solutions, we not only strengthened our confidence in the correctness of our answers but also deepened our understanding of the underlying concepts. The solutions k = 2/3 and k = -1 represent specific conditions under which the given quadratic equation exhibits a unique characteristic – equal roots. This underscores the power of algebraic techniques in uncovering hidden properties and relationships within mathematical expressions.