Finding The Value Of K In Quadratic Equation 2x² - 12x + K + 2 = 0 When Roots Differ By 2
In this comprehensive guide, we will explore how to determine the value of k in a quadratic equation where the roots differ by 2. We will delve into the fundamental concepts of quadratic equations, their roots, and the relationships between the coefficients and roots. By applying these principles, we will develop a step-by-step approach to solve this problem. This exploration is crucial for anyone studying algebra, especially quadratic equations and their properties.
Understanding Quadratic Equations and Their Roots
At the heart of this problem lies the understanding of quadratic equations. 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 ≠ 0. The roots of a quadratic equation are the values of x that satisfy the equation, or in other words, the points where the parabola represented by the equation intersects the x-axis. These roots are also known as solutions or zeros of the equation. Understanding these roots and how they relate to the equation's coefficients is fundamental to solving problems like finding the value of k when the roots have a specific relationship.
The roots of a quadratic equation can be found using various methods, including factoring, completing the square, and the quadratic formula. The quadratic formula, a cornerstone in solving quadratic equations, is given by:
x = (-b ± √(b² - 4ac)) / (2a)
This formula provides a direct way to calculate the roots, denoted as x₁ and x₂, based on the coefficients a, b, and c. The expression inside the square root, b² - 4ac, is known as the discriminant, which provides valuable information about the nature of the roots. If the discriminant is positive, the equation has two distinct real roots; if it is zero, the equation has one real root (a repeated root); and if it is negative, the equation has two complex roots. This understanding of the discriminant is vital in predicting the type of solutions we will encounter.
Furthermore, there exist relationships between the roots and the coefficients of the quadratic equation. For a quadratic equation ax² + bx + c = 0, the sum of the roots (x₁ + x₂) is equal to -b/a, and the product of the roots (x₁ * x*₂) is equal to c/a. These relationships, often referred to as Vieta's formulas, provide a powerful tool for solving problems involving roots without explicitly calculating them. They allow us to connect the roots' sum and product directly to the coefficients of the quadratic equation, which is particularly useful when given information about the roots' relationship, as in our problem where the roots differ by 2. By leveraging these relationships, we can formulate equations involving k and solve for its value.
Problem Statement: 2x² - 12x + k + 2 = 0
Now, let's focus on the specific problem at hand. We are given the quadratic equation 2x² - 12x + k + 2 = 0, and we know that the roots of this equation differ by 2. Our goal is to find the value of k. This problem is a classic example of how the relationships between the roots and coefficients of a quadratic equation can be used to solve for unknown parameters. The key to solving this problem lies in translating the given condition (roots differing by 2) into a mathematical equation and then using Vieta's formulas to relate this equation to the coefficients of the given quadratic equation.
To begin, let's denote the roots of the equation as α and β. According to the problem statement, the difference between the roots is 2, which can be expressed as |α - β| = 2. This condition provides us with a crucial piece of information that we can use in conjunction with Vieta's formulas. Vieta's formulas, as mentioned earlier, state that for a quadratic equation ax² + bx + c = 0, the sum of the roots (α + β) is -b/a, and the product of the roots (αβ) is c/a. In our case, a = 2, b = -12, and c = k + 2. Therefore, we can express the sum and product of the roots in terms of these coefficients.
Applying Vieta's formulas to our equation, we have:
α + β = -(-12) / 2 = 6
αβ = (k + 2) / 2
We now have two equations: |α - β| = 2 and α + β = 6. The absolute value in the first equation suggests that we might need to consider two cases: α - β = 2 and α - β = -2. However, since we are ultimately interested in the value of k, and k is related to the product of the roots, the sign difference will not affect our final answer. We can proceed with either case, and the result will be the same. Let's proceed with α - β = 2. Now we have a system of two linear equations with two unknowns, α and β, which we can solve using various methods, such as substitution or elimination.
Step-by-Step Solution
Let's proceed with solving the system of equations. We have:
α + β = 6
α - β = 2
We can solve this system by adding the two equations together. This will eliminate β and allow us to solve for α. Adding the equations, we get:
2α = 8
Dividing both sides by 2, we find:
α = 4
Now that we have the value of α, we can substitute it back into either of the original equations to find β. Let's use the first equation, α + β = 6:
4 + β = 6
Subtracting 4 from both sides, we get:
β = 2
So, the roots of the equation are α = 4 and β = 2. This confirms that the roots indeed differ by 2, as stated in the problem. Now that we have the values of α and β, we can use the relationship for the product of the roots to find the value of k. We know that:
αβ = (k + 2) / 2
Substituting the values of α and β, we get:
4 * 2 = (k + 2) / 2
8 = (k + 2) / 2
Multiplying both sides by 2, we get:
16 = k + 2
Subtracting 2 from both sides, we find:
k = 14
Therefore, the value of k that satisfies the given condition is 14. This completes the solution to the problem. We have successfully used the relationships between the roots and coefficients of a quadratic equation, along with the given condition about the difference between the roots, to find the unknown parameter k. This solution demonstrates the power and elegance of Vieta's formulas in solving problems involving quadratic equations.
Verification of the Solution
To ensure the accuracy of our solution, it is always a good practice to verify the result. We found that k = 14, so let's substitute this value back into the original equation and see if the roots indeed differ by 2. The equation becomes:
2x² - 12x + 14 + 2 = 0
2x² - 12x + 16 = 0
We can simplify this equation by dividing all terms by 2:
x² - 6x + 8 = 0
Now, we can factor this quadratic equation. We are looking for two numbers that multiply to 8 and add up to -6. These numbers are -4 and -2. So, we can factor the equation as:
(x - 4)(x - 2) = 0
This gives us the roots:
x₁ = 4
x₂ = 2
The roots are indeed 4 and 2, which differ by 2, as required by the problem statement. This verification confirms that our solution for k is correct. The process of verification not only gives us confidence in our solution but also reinforces our understanding of the concepts and techniques used in solving the problem. It highlights the importance of careful and methodical problem-solving in mathematics.
Alternative Approaches
While we solved the problem using Vieta's formulas and a system of equations, there are alternative approaches that can be used to arrive at the same solution. One such approach involves using the quadratic formula directly in conjunction with the given condition about the difference between the roots. This method can provide a different perspective on the problem and further enhance our understanding of quadratic equations.
Recall the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
For our equation, 2x² - 12x + k + 2 = 0, we have a = 2, b = -12, and c = k + 2. Let's denote the roots as x₁ and x₂:
x₁ = (12 + √(12² - 4 * 2 * (k + 2))) / (2 * 2)
x₂ = (12 - √(12² - 4 * 2 * (k + 2))) / (2 * 2)
We know that the roots differ by 2, so |x₁ - x₂| = 2. Substituting the expressions for x₁ and x₂, we get:
|(12 + √(144 - 8(k + 2))) / 4 - (12 - √(144 - 8(k + 2))) / 4| = 2
Simplifying the expression inside the absolute value, we get:
|(2√(144 - 8k - 16)) / 4| = 2
|√(128 - 8k) / 2| = 2
Squaring both sides, we get:
(128 - 8k) / 4 = 4
128 - 8k = 16
8k = 112
k = 14
This alternative approach, using the quadratic formula directly, yields the same result, k = 14. This demonstrates that there can be multiple ways to solve a mathematical problem, and choosing the most efficient method often depends on the specific problem and one's familiarity with different techniques. This approach also highlights the importance of algebraic manipulation and simplification in solving equations.
Conclusion
In conclusion, we have successfully found the value of k in the quadratic equation 2x² - 12x + k + 2 = 0, given that the roots differ by 2. We explored the fundamental concepts of quadratic equations, their roots, and the relationships between coefficients and roots, particularly Vieta's formulas. We developed a step-by-step approach to solve the problem, which involved translating the given condition into a mathematical equation, using Vieta's formulas to relate this equation to the coefficients, and solving for k. We also verified our solution and explored an alternative approach using the quadratic formula directly. This comprehensive exploration provides a solid understanding of how to solve this type of problem and reinforces the importance of understanding the underlying principles of quadratic equations. Mastering these concepts is essential for success in algebra and related fields. The ability to connect the roots of a quadratic equation to its coefficients is a powerful tool in mathematical problem-solving. By understanding and applying these relationships, we can solve a wide range of problems involving quadratic equations and their properties. This problem serves as an excellent example of how theoretical knowledge can be applied to solve practical problems, and it encourages a deeper understanding of mathematical concepts.
