Solving quadratic equations is a fundamental aspect of algebra, and understanding the role of specific coefficients, such as k, is crucial. In this article, we delve into the equation X2 6X k 0 to explore how the value of k impacts the nature of the roots.

Introduction to Quadratic Equations

A quadratic equation is any equation that can be written in the form ax2 bx c 0. The term a must be non-zero. In our specific example, a 1, b 6, and c k. The roots of this equation can provide valuable insights into the behavior of the quadratic function represented by the equation.

The Role of k in the Equation

To determine the value of k that results in real roots, we can use the discriminant. The discriminant, denoted by D, helps us understand the nature of the roots of the quadratic equation and is given by the formula:

D b2 - 4ac

For the equation X2 6X k 0, we substitute a 1, b 6, and c k. This gives us:

D 62 - 4(1)(k) 36 - 4k

The roots of the equation will be real if the discriminant is greater than or equal to zero:

36 - 4k ≥ 0

Solving for k involves simple algebraic manipulation:

36 ≥ 4k

9 ≥ k

Therefore, the value of k must be less than or equal to 9 for the quadratic equation to have real roots. This condition is a critical determining factor.

Sum and Product of Roots

For a quadratic equation of the form ax2 bx c 0, the sum of the roots is given by:

-b/a

and the product of the roots is given by:

c/a

Given a 1, b 6, and c k, the sum and product of the roots are:

Sum of roots -6

Product of roots k

The pairs of roots that satisfy these conditions can be found as follows:

-1, -5 such that -1 * -5 5

0, -6 such that 0 * -6 0

-2, -3 such that -2 * -3 6

-3, -3 such that -3 * -3 9

The corresponding values of k are:

k 5, 0, 6, 9

Step-by-Step Solution

To solve the quadratic equation X2 6X k 0 explicitly, we can use the quadratic formula:

X (-b ± sqrt( b2 - 4ac )) / (2a)

For our equation, we get:

X (-6 ± sqrt( 62 - 4*1*k )) / 2

Simplifying:

X -3 ± sqrt( 36 - 4k )

Thus, the solutions for X are:

X -3 sqrt(9 - k) and X -3 - sqrt(9 - k)

These solutions are valid for k ≤ 9

Alternative Approach

Another method to solve the quadratic equation involves completing the square. Starting with the equation:

X2 6X k 0

We can rewrite it as:

X2 6X -k

Adding 9 to both sides:

X2 6X 9 -k 9

This can be written as:

(X 3)2 9 - k

Thus:

X 3 ±sqrt(9 - k)

Finally, we get:

X -3 ± sqrt(9 - k)

These are the roots of the equation for k ≤ 9.

Conclusion

In conclusion, the value of k in the equation X2 6X k 0 must be less than or equal to 9 for the equation to have real roots. The roots are determined by the discriminant and can also be found using the quadratic formula or by completing the square. Understanding these methods is essential for solving quadratic equations and has wide applications in mathematics and other sciences.