Solving Quadratic Equations: How to Find the Value of k Given a Root

In algebra, quadratic equations are fundamental for understanding various mathematical concepts. One common problem is determining the value of a constant given a root of the equation. This article will walk through a detailed solution for the equation x^2 - 4xk 0 with a root of 6.

Understanding the Problem

We are given the quadratic equation in the form x^2 - 4xk 0. The problem states that one of the roots of this equation is 6. Our objective is to find the value of the constant k.

Step-by-Step Solution

To solve this, we can follow these steps:

Substitute x 6 into the equation x^2 - 4xk 0. Calculate and simplify the equation step by step. Solve for k.

Step 1: Substitution

Substituting x 6 into the equation:

6^2 - 4(6)k 0

Simplifying, we get:

36 - 24k 0

Step 2: Simplification

From the above equation, isolate 24k:

36 24k

Simplifying further, we have:

12k 0

Step 3: Solving for k

Solve for k:

k -12

Thus, the value of k is boxed{-12}.

Alternative Solutions

There are several ways to arrive at the same solution. Here, we present alternative methods:

Method 1: Completing the Square

We can rewrite the equation:

x^2 - 4xk 0

Using the roots form, we have:

x^2 - 4xk (x - 6)(x - b) 0

Comparing both sides, we get:

-4 -6 - b

Thus, b -2

Substituting b back in the equation:

k 6b -12

Method 2: Direct Substitution and Solving

Directly substituting x 6 and solving:

(6)^2 - 4(6)k 0

36 - 24k 0

12k 36

k -12

Enhanced Understanding

By understanding the methods and the underlying principles, one can solve similar problems more efficiently. The value of k in the equation x^2 - 4xk 0, given that one of the roots is 6, is -12.

Conclusion

Through these solutions, we have demonstrated how to find the value of k in a quadratic equation when given a root. This method can be applied to more complex equations as well. Understanding the algebraic steps and methods will help in solving such problems effectively.