Finding Coefficients P And Q Given Roots Of Quadratic Equation
In this article, we delve into the fascinating world of quadratic equations and explore how to determine the coefficients of a quadratic equation when its roots are known. Specifically, we will tackle the problem of finding the values of p and q in the quadratic equation px² + 7x + q = 0, given that its roots are 2/3 and -3. This problem beautifully illustrates the relationship between the roots and coefficients of a quadratic equation, a fundamental concept in algebra. Understanding this relationship not only helps in solving such problems but also provides a deeper insight into the structure and behavior of quadratic equations.
Before diving into the solution, let's establish a solid understanding of the key concepts involved. A quadratic equation is a polynomial equation of the second degree, generally represented in the form ax² + 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, meaning that when these values are substituted into the equation, the equation holds true. A quadratic equation has exactly two roots, which may be real or complex, and they may be distinct or repeated.
The relationship between the roots and coefficients of a quadratic equation is a cornerstone concept. For a quadratic equation ax² + bx + c = 0, if the roots are denoted as α and β, then the sum of the roots (α + β) is equal to -b/a, and the product of the roots (αβ) is equal to c/a. These relationships provide a direct link between the coefficients of the equation and its solutions, allowing us to determine the equation if the roots are known, and vice versa. These relationships are derived from Vieta's formulas, which generalize these relationships for polynomials of any degree.
In our specific problem, we are given the quadratic equation px² + 7x + q = 0 and its roots 2/3 and -3. Our goal is to find the values of p and q. To achieve this, we will utilize the relationships between the roots and coefficients, applying the sum and product formulas to set up a system of equations that can be solved for p and q. This process involves careful algebraic manipulation and a clear understanding of the underlying principles.
To find the values of p and q, we will leverage the fundamental relationships between the roots and the coefficients of a quadratic equation. Given the roots 2/3 and -3 of the equation px² + 7x + q = 0, we can apply the formulas for the sum and product of the roots. Let's denote the roots as α = 2/3 and β = -3. The sum of the roots, α + β, is equal to 2/3 + (-3) = 2/3 - 9/3 = -7/3. According to the root-coefficient relationship, the sum of the roots is also equal to -b/a, where a is the coefficient of x² and b is the coefficient of x. In our equation, a = p and b = 7, so we have the equation -7/p = -7/3.
The product of the roots, αβ, is equal to (2/3) * (-3) = -2. Similarly, the product of the roots is also equal to c/a, where c is the constant term. In our equation, c = q, so we have the equation q/p = -2. Now, we have a system of two equations:
- -7/p = -7/3
- q/p = -2
Solving this system will give us the values of p and q. The first equation allows us to directly solve for p. The second equation then allows us to find q once we know p. This methodical approach ensures that we can accurately determine the coefficients of the quadratic equation based on its given roots. By carefully applying these relationships and performing the necessary algebraic steps, we can successfully solve the problem.
Now that we have established the equations, let's proceed with solving for p and q. From the first equation, -7/p = -7/3, we can solve for p by cross-multiplying or simply observing that the numerators are equal, which implies that the denominators must also be equal. Thus, we have p = 3. This value of p is crucial as it forms the foundation for finding q.
Next, we substitute the value of p into the second equation, q/p = -2. Replacing p with 3, we get q/3 = -2. To solve for q, we multiply both sides of the equation by 3, which gives us q = -2 * 3 = -6. Therefore, the value of q is -6. With both p and q determined, we have successfully found the coefficients of the quadratic equation.
To verify our solution, we can substitute the values of p and q back into the original equation and check if the given roots satisfy the equation. The equation becomes 3x² + 7x - 6 = 0. Substituting x = 2/3, we get 3(2/3)² + 7(2/3) - 6 = 3(4/9) + 14/3 - 6 = 4/3 + 14/3 - 18/3 = 0, which confirms that 2/3 is indeed a root. Similarly, substituting x = -3, we get 3(-3)² + 7(-3) - 6 = 3(9) - 21 - 6 = 27 - 21 - 6 = 0, which confirms that -3 is also a root. This verification step is essential to ensure the accuracy of our solution and to reinforce our understanding of the relationship between roots and coefficients.
In conclusion, we have successfully determined the values of p and q in the quadratic equation px² + 7x + q = 0, given that its roots are 2/3 and -3. By applying the relationships between the roots and coefficients of a quadratic equation, we established a system of equations and solved for p and q. We found that p = 3 and q = -6. This problem highlights the power of algebraic principles in solving mathematical challenges and underscores the importance of understanding fundamental concepts in quadratic equations.
The process of finding coefficients from roots is a valuable skill in algebra and has applications in various fields, including engineering, physics, and computer science. The ability to manipulate equations and apply mathematical relationships is crucial for problem-solving in these disciplines. Moreover, this exercise reinforces the understanding of how the roots of a polynomial equation are intrinsically linked to its coefficients, a connection that is central to the study of polynomials and their properties. By mastering these concepts, students can develop a deeper appreciation for the elegance and utility of mathematics.
Find the values of p and q if 2/3 and -3 are the roots of the equation px² + 7x + q = 0.
