If 4 Lies Between The Roots Of The Equation X^2 - (3k-1)x + 5k = 0, What Is The Minimum Possible Integral Value Of K?

In the realm of quadratic equations, a fascinating problem arises when we explore the relationship between the roots of an equation and a given number. This article delves into such a problem, specifically focusing on the equation x^2 - (3k-1)x + 5k = 0 and the condition that the number '4' lies between its roots. Our primary objective is to determine the minimum possible integral value of 'k' that satisfies this condition. This exploration will not only enhance our understanding of quadratic equations but also showcase the power of analytical techniques in solving mathematical puzzles.

Problem Statement: Decoding the Roots and the Intervening Number

Let's begin by restating the problem clearly. We are given the quadratic equation x^2 - (3k-1)x + 5k = 0. The core challenge lies in finding the minimum integral value of 'k' such that the number '4' lies strictly between the two roots of this equation. To tackle this, we need to understand the implications of a number lying between the roots of a quadratic equation. This condition provides us with crucial information about the sign of the quadratic expression when evaluated at that number. By leveraging this information, we can establish an inequality that will help us determine the possible values of 'k'. This problem is a beautiful blend of algebraic manipulation and insightful reasoning, which makes it an excellent exercise for anyone interested in deepening their mathematical prowess.

Laying the Foundation: The Significance of '4' Between the Roots

To solve this problem effectively, we need to understand a fundamental concept in quadratic equations. When a number lies between the roots of a quadratic equation, it implies that the quadratic expression changes its sign at that number. In simpler terms, if '4' lies between the roots of our equation x^2 - (3k-1)x + 5k = 0, then substituting x = 4 into the equation will result in a value with the opposite sign of the leading coefficient (which is 1 in this case, a positive value). Therefore, the result of the substitution must be negative. This is a critical insight that forms the basis of our solution. We will use this concept to formulate an inequality involving 'k', which we can then solve to find the range of possible values for 'k'. This step is crucial as it translates the geometric condition (4 lying between the roots) into an algebraic condition that we can work with.

The Inequality Unveiled: Substituting and Simplifying

Now, let's put our understanding into action. We substitute x = 4 into the quadratic expression x^2 - (3k-1)x + 5k and set the result to be less than zero, as discussed earlier. This gives us the inequality: (4)^2 - (3k-1)(4) + 5k < 0. Simplifying this inequality is the next step. Expanding the terms, we get 16 - 12k + 4 + 5k < 0. Combining like terms, we arrive at a simpler inequality: 20 - 7k < 0. This inequality is a crucial stepping stone in our solution. It directly relates 'k' to the condition that '4' lies between the roots of the equation. Our next task is to solve this inequality for 'k', which will give us the range of values that 'k' can take while satisfying the given condition. This process demonstrates the power of algebraic manipulation in transforming a problem into a solvable form.

Solving for k: The Range of Possibilities

With the inequality 20 - 7k < 0 in hand, we can now solve for 'k'. Adding 7k to both sides, we get 20 < 7k. Dividing both sides by 7, we find k > 20/7. Since 20/7 is approximately 2.86, this inequality tells us that 'k' must be greater than 2.86. However, we are looking for the minimum possible integral value of 'k'. This means we need to find the smallest integer that satisfies this inequality. The smallest integer greater than 2.86 is 3. Therefore, the minimum possible integral value of 'k' is 3. This result is a testament to the precision of mathematical reasoning. By carefully applying the principles of quadratic equations and inequalities, we have successfully pinpointed the exact value of 'k' that meets the given criteria.

Conclusion: The Triumph of Analytical Thinking

In conclusion, we have successfully determined that the minimum possible integral value of 'k' for the equation x^2 - (3k-1)x + 5k = 0, such that '4' lies between its roots, is 3. This problem showcases the elegance and power of mathematical problem-solving. By understanding the relationship between the roots of a quadratic equation and the sign of the quadratic expression, we were able to formulate an inequality and solve for 'k'. This journey from the initial problem statement to the final solution highlights the importance of clear thinking, logical reasoning, and algebraic manipulation in the world of mathematics. Problems like this not only sharpen our mathematical skills but also instill a sense of accomplishment in unraveling complex relationships and arriving at precise answers. The beauty of mathematics lies in its ability to provide definitive solutions, and this problem serves as a perfect example of that.