Finding The Value Of K For Equal Roots In Quadratic Equations
In mathematics, understanding the nature of roots in a quadratic equation is a fundamental concept. A quadratic equation, typically expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠0, has roots that can be real or complex, and these roots determine the points where the parabola intersects the x-axis. The discriminant, denoted as Δ, plays a crucial role in determining the nature of these roots. The discriminant is given by the formula Δ = b² - 4ac. If Δ > 0, the equation has two distinct real roots; if Δ = 0, the equation has exactly one real root (or two equal real roots); and if Δ < 0, the equation has two complex roots. This article delves into the specific problem of finding the value of k for which the quadratic equation x² + 2(k + 1)x + k² = 0 has equal roots. This involves understanding the condition for equal roots and applying it to the given equation to solve for k. By setting the discriminant to zero, we can find the values of k that satisfy the condition for equal roots. This exploration not only reinforces the understanding of quadratic equations but also highlights the application of discriminant in solving such problems.
To understand the value of k for which the given equation has equal roots, it's essential to revisit the basics of quadratic equations and their roots. A quadratic equation is generally represented as ax² + bx + c = 0, where a, b, and c are coefficients, and x is the variable. The roots of a quadratic equation are the values of x that satisfy the equation, i.e., the values of x for which the equation equals zero. These roots can be real or complex, and their nature is determined by the discriminant (Δ). The discriminant is a part of the quadratic formula, which is used to find the roots of a quadratic equation. The quadratic formula is given by:
x = [-b ± √(b² - 4ac)] / (2a)
Within this formula, the expression b² - 4ac is the discriminant (Δ). The discriminant provides valuable information about the nature of the roots: if Δ > 0, the equation has two distinct real roots; if Δ = 0, the equation has exactly one real root (or two equal real roots); and if Δ < 0, the equation has two complex roots. The case where Δ = 0 is of particular interest when seeking equal roots. Equal roots mean that the quadratic equation touches the x-axis at only one point, indicating that the vertex of the parabola lies on the x-axis. In practical terms, this means that the quadratic equation has a repeated root. Understanding these fundamental concepts is crucial for solving problems related to quadratic equations and their roots, including the problem at hand, where we need to find the value of k for which the equation x² + 2(k + 1)x + k² = 0 has equal roots.
In the realm of quadratic equations, the condition for equal roots is a critical concept that arises when the discriminant (Δ) of the quadratic equation is equal to zero. The discriminant, as mentioned earlier, is given by the formula Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. When Δ = 0, it implies that the quadratic equation has exactly one real root, or two equal real roots. This situation occurs because the square root term in the quadratic formula, √(b² - 4ac), becomes zero, resulting in the formula simplifying to x = -b/(2a). This single root represents the x-coordinate where the parabola, which is the graphical representation of the quadratic equation, touches the x-axis at only one point. Geometrically, this means that the vertex of the parabola lies on the x-axis. The condition Δ = 0 is not just a theoretical concept but a practical tool for solving problems where equal roots are required. In the context of our problem, where we are given the quadratic equation x² + 2(k + 1)x + k² = 0, we will use this condition to find the value of k for which the equation has equal roots. By setting the discriminant of this equation to zero, we can derive an equation in terms of k and solve for it. This process demonstrates the direct application of the condition for equal roots in solving quadratic equation problems.
To apply the discriminant to the given equation x² + 2(k + 1)x + k² = 0, we first need to identify the coefficients a, b, and c. In this equation, a = 1 (the coefficient of x²), b = 2(k + 1) (the coefficient of x), and c = k² (the constant term). The discriminant, Δ, is given by the formula Δ = b² - 4ac. Substituting the values of a, b, and c into this formula, we get:
Δ = [2(k + 1)]² - 4(1)(k²)
Expanding and simplifying this expression, we have:
Δ = 4(k² + 2k + 1) - 4k² Δ = 4k² + 8k + 4 - 4k² Δ = 8k + 4
Now, for the equation to have equal roots, the discriminant must be equal to zero. Therefore, we set Δ = 0 and solve for k:
8k + 4 = 0 8k = -4 k = -4/8 k = -1/2
Thus, the value of k for which the given equation has equal roots is -1/2. This process demonstrates the practical application of the discriminant in determining the nature of roots and solving for unknown parameters in a quadratic equation. By correctly identifying the coefficients and applying the discriminant formula, we were able to find the specific value of k that satisfies the condition for equal roots.
Solving for k involves setting the discriminant equal to zero and finding the value of k that satisfies this condition. As we derived in the previous section, the discriminant for the given equation x² + 2(k + 1)x + k² = 0 is Δ = 8k + 4. For equal roots, we need Δ = 0. So, we set up the equation:
8k + 4 = 0
To solve for k, we first subtract 4 from both sides of the equation:
8k = -4
Next, we divide both sides by 8:
k = -4/8
Simplifying the fraction, we get:
k = -1/2
Therefore, the value of k for which the equation x² + 2(k + 1)x + k² = 0 has equal roots is -1/2. This solution is obtained by directly applying the condition for equal roots (Δ = 0) and solving the resulting linear equation in k. The steps involved are straightforward algebraic manipulations, making it a clear and concise method for finding the desired value of k. This result confirms that when k = -1/2, the given quadratic equation will have exactly one real root, which is a repeated root.
In conclusion, the value of k for which the equation x² + 2(k + 1)x + k² = 0 has equal roots is -1/2. This result was obtained by applying the fundamental concept of the discriminant in quadratic equations. The discriminant, given by the formula Δ = b² - 4ac, is a crucial tool for determining the nature of roots in a quadratic equation. When the discriminant is equal to zero, the equation has exactly one real root, or two equal real roots. By identifying the coefficients a, b, and c in the given equation, calculating the discriminant, and setting it to zero, we formed a linear equation in terms of k. Solving this equation yielded k = -1/2, which is the value that satisfies the condition for equal roots. This exercise underscores the importance of understanding the discriminant and its applications in solving quadratic equation problems. The ability to determine the nature of roots and find unknown parameters based on root conditions is a valuable skill in algebra and mathematical problem-solving. The process demonstrated here provides a clear and systematic approach to solving such problems, reinforcing the connection between theoretical concepts and practical applications in mathematics.
