Finding Values Of P For Equal Roots In A Quadratic Equation
In the realm of mathematics, quadratic equations hold a significant place. These equations, characterized by their highest power of 2, manifest in various forms and applications. One intriguing aspect of quadratic equations lies in the nature of their roots – the values of the variable that satisfy the equation. Roots can be real or complex, distinct or repeated. In this comprehensive article, we delve into a specific scenario: determining the values of a parameter, denoted as 'p', for which a given quadratic equation possesses equal roots. This exploration will involve a detailed analysis of the discriminant, a crucial component that unveils the nature of the roots, and the application of algebraic techniques to solve for the desired values of 'p'. This analysis not only reinforces our understanding of quadratic equations but also highlights the power of mathematical tools in solving specific problems.
The journey begins with a clear statement of the problem at hand: to find the values of 'p' that ensure the quadratic equation (2p + 1)x^2 - (7p + 2)x + (7p - 3) = 0 has equal roots. The first step in this endeavor is to understand the significance of equal roots. Equal roots, also known as repeated roots, occur when the quadratic equation has exactly one solution. This condition is intimately linked to the discriminant of the quadratic equation, which we will explore in detail in the subsequent sections. The discriminant, denoted by the Greek letter delta (Δ), is a mathematical expression derived from the coefficients of the quadratic equation. Its value provides critical information about the nature of the roots: a positive discriminant indicates two distinct real roots, a zero discriminant indicates equal real roots, and a negative discriminant indicates complex roots. Therefore, to achieve equal roots, we must ensure that the discriminant of the given quadratic equation is equal to zero. This sets the stage for a series of algebraic manipulations and problem-solving strategies aimed at isolating and determining the values of 'p' that satisfy this condition. As we proceed, we will encounter and apply fundamental concepts of algebra, including the quadratic formula, factoring, and equation solving, reinforcing the interconnectedness of mathematical ideas.
Understanding the Discriminant
To understand the discriminant better, we must first revisit the general form of a quadratic equation. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are coefficients, and x is the variable. The discriminant, denoted by Δ (delta), is a mathematical expression derived from these coefficients and is defined as Δ = b^2 - 4ac. This simple yet powerful formula holds the key to unlocking the nature of the roots of the quadratic equation. The discriminant's value provides a concise summary of the roots' characteristics, allowing us to predict whether they are real or complex, distinct or repeated. When the discriminant is positive (Δ > 0), the quadratic equation has two distinct real roots. This means that there are two different values of x that satisfy the equation, and these values are real numbers. Geometrically, this corresponds to the parabola represented by the quadratic equation intersecting the x-axis at two distinct points. In contrast, when the discriminant is negative (Δ < 0), the quadratic equation has two complex roots. Complex roots involve imaginary numbers, which are multiples of the imaginary unit 'i', defined as the square root of -1. Complex roots arise when the quadratic equation's parabola does not intersect the x-axis. The solutions are expressed in the form a + bi and a - bi, where a and b are real numbers, and 'i' is the imaginary unit. The crucial case for our problem is when the discriminant is equal to zero (Δ = 0). In this scenario, the quadratic equation has exactly one real root, which is considered a repeated root or equal roots. This means that the parabola touches the x-axis at a single point, representing the only value of x that satisfies the equation. This condition of equal roots is the central focus of our investigation, as we aim to find the values of 'p' that make the discriminant of the given quadratic equation equal to zero.
Applying the Discriminant to the Given Equation
Now, let's apply the discriminant concept to the given quadratic equation: (2p + 1)x^2 - (7p + 2)x + (7p - 3) = 0. To find the values of 'p' for which this equation has equal roots, we need to first identify the coefficients a, b, and c in terms of 'p'. Comparing the given equation with the general form ax^2 + bx + c = 0, we can identify the coefficients as follows: a = (2p + 1), b = -(7p + 2), and c = (7p - 3). These coefficients, expressed in terms of 'p', will be crucial in calculating the discriminant. Recall that the discriminant Δ is given by the formula Δ = b^2 - 4ac. To ensure the quadratic equation has equal roots, we need to set the discriminant equal to zero (Δ = 0). Substituting the identified coefficients into the discriminant formula, we get the equation: (-(7p + 2))^2 - 4(2p + 1)(7p - 3) = 0. This equation forms the core of our problem-solving process. It is a quadratic equation in 'p', and solving it will reveal the values of 'p' that satisfy the condition of equal roots. The next step involves expanding and simplifying this equation. We need to carefully expand the squared term and the product of the two binomials, paying close attention to the signs and distribution. The expansion process will lead to a polynomial expression in 'p', which we will then simplify by combining like terms. The resulting equation will be a quadratic equation in the standard form, which we can then solve using various methods, such as factoring, completing the square, or the quadratic formula. The solutions for 'p' obtained from this equation will be the values that make the discriminant zero, thus ensuring that the original quadratic equation has equal roots. This meticulous process of substituting, expanding, and simplifying is essential to arrive at the correct solutions and demonstrate the power of algebraic manipulation in solving mathematical problems.
Solving for p
With the discriminant equation established, the next step is to solve for p. Expanding the equation (-(7p + 2))^2 - 4(2p + 1)(7p - 3) = 0, we get: (49p^2 + 28p + 4) - 4(14p^2 - 6p + 7p - 3) = 0. Now, we simplify the equation by distributing the -4 and combining like terms: 49p^2 + 28p + 4 - 56p^2 + 24p - 28p + 12 = 0. Combining the p^2 terms, the p terms, and the constant terms, we get: -7p^2 + 24p + 16 = 0. To make the equation easier to work with, we can multiply the entire equation by -1, resulting in: 7p^2 - 24p - 16 = 0. This is a standard quadratic equation in the form ap^2 + bp + c = 0, where a = 7, b = -24, and c = -16. To solve this quadratic equation, we can use the quadratic formula, which states that for an equation of the form ap^2 + bp + c = 0, the solutions for p are given by: p = (-b ± √(b^2 - 4ac)) / (2a). Substituting the values of a, b, and c into the quadratic formula, we get: p = (24 ± √((-24)^2 - 4 * 7 * (-16))) / (2 * 7). Simplifying the expression under the square root: p = (24 ± √(576 + 448)) / 14. Further simplification yields: p = (24 ± √1024) / 14. Since √1024 = 32, we have: p = (24 ± 32) / 14. Now, we have two possible solutions for p: p1 = (24 + 32) / 14 = 56 / 14 = 4, and p2 = (24 - 32) / 14 = -8 / 14 = -4 / 7. Therefore, the values of p for which the given quadratic equation has equal roots are p = 4 and p = -4/7. These values represent the specific parameter settings that cause the discriminant of the equation to be zero, ensuring that the equation has exactly one real solution. This process demonstrates the practical application of the discriminant and the quadratic formula in solving problems involving quadratic equations.
Verifying the Solutions
Having obtained the solutions for p, it is crucial to verify their correctness. This verification process involves substituting each value of p back into the original quadratic equation and checking if the resulting equation indeed has equal roots. This step not only ensures the accuracy of our calculations but also reinforces our understanding of the relationship between the parameter 'p' and the nature of the roots. Let's start by substituting p = 4 into the original equation: (2p + 1)x^2 - (7p + 2)x + (7p - 3) = 0. Substituting p = 4, we get: (2(4) + 1)x^2 - (7(4) + 2)x + (7(4) - 3) = 0, which simplifies to: 9x^2 - 30x + 25 = 0. This is a quadratic equation, and we can check if it has equal roots by calculating its discriminant. The discriminant Δ is given by b^2 - 4ac, where a = 9, b = -30, and c = 25. Calculating the discriminant, we get: Δ = (-30)^2 - 4 * 9 * 25 = 900 - 900 = 0. Since the discriminant is zero, the quadratic equation has equal roots, confirming that p = 4 is a valid solution. Next, let's substitute p = -4/7 into the original equation: (2p + 1)x^2 - (7p + 2)x + (7p - 3) = 0. Substituting p = -4/7, we get: (2(-4/7) + 1)x^2 - (7(-4/7) + 2)x + (7(-4/7) - 3) = 0, which simplifies to: (-1/7)x^2 + 2x - 7 = 0. To eliminate the fraction, we can multiply the entire equation by -7, resulting in: x^2 - 14x + 49 = 0. Again, we calculate the discriminant to check for equal roots. The discriminant Δ is given by b^2 - 4ac, where a = 1, b = -14, and c = 49. Calculating the discriminant, we get: Δ = (-14)^2 - 4 * 1 * 49 = 196 - 196 = 0. Since the discriminant is zero, the quadratic equation has equal roots, confirming that p = -4/7 is also a valid solution. Therefore, we have successfully verified that both p = 4 and p = -4/7 are the values of p for which the given quadratic equation has equal roots. This verification process underscores the importance of checking solutions in mathematical problem-solving, ensuring accuracy and a deeper understanding of the concepts involved.
Conclusion
In conclusion, we have successfully determined the values of 'p' for which the quadratic equation (2p + 1)x^2 - (7p + 2)x + (7p - 3) = 0 has equal roots. By leveraging the concept of the discriminant, we established the condition for equal roots: the discriminant must be equal to zero. We then applied this condition to the given equation, carefully substituting the coefficients and simplifying the resulting equation in terms of 'p'. This led us to a quadratic equation in 'p', which we solved using the quadratic formula. The solutions obtained were p = 4 and p = -4/7. To ensure the accuracy of our solutions, we performed a verification process, substituting each value of 'p' back into the original equation and confirming that the resulting quadratic equations indeed had equal roots. This comprehensive approach not only provided the correct answers but also reinforced the importance of understanding the underlying mathematical principles and the need for thoroughness in problem-solving. The discriminant, as a tool for analyzing the nature of roots, plays a crucial role in the study of quadratic equations. Its application extends beyond this specific problem, serving as a fundamental concept in various mathematical contexts. The ability to manipulate algebraic expressions, solve quadratic equations, and verify solutions are essential skills in mathematics, and this exploration has provided a valuable opportunity to practice and enhance these skills. This journey through the realm of quadratic equations and their roots highlights the interconnectedness of mathematical concepts and the power of algebraic techniques in solving real-world problems.
