Determining the Value of k for Quadratic Equations with Equal Roots

In this article, we will explore how to determine the value of the coefficient k for a quadratic equation that results in two equal roots. We will delve into the concept of the discriminant, which is crucial in identifying whether a quadratic equation has two distinct real roots, one real root, or no real roots. We will analyze specific examples to solidify our understanding and provide a clear guide to solving such problems.

Understanding the Discriminant

The discriminant, denoted as ( D ) or ( b^2 - 4ac ), determines the nature of the roots of a quadratic equation. The formula for the quadratic equation is given by ( ax^2 bx c 0 ), where ( a ), ( b ), and ( c ) are coefficients.

Equality of Roots

If a quadratic equation has two equal roots, then the discriminant must equal zero. This happens because the quadratic expression forms a perfect square, leading to a single, repeated root. Mathematically, this can be expressed as:

[D 0 quad Rightarrow quad b^2 - 4ac 0]

Example 1: Quadratic Equation with Equal Roots

Consider the quadratic equation ( kx^2 - 4x 1 0 ). For this equation to have two equal roots, the discriminant must be zero. Thus:

[D (-4)^2 - 4(k)(1) 0]

Let's solve for k:

[16 - 4k 0 quad Rightarrow quad 4k 16 quad Rightarrow quad k 4]

Upon verification, substituting k 4 in the original equation:

[4x^2 - 4x 1 0]

We can factorize this as:

[(2x - 1)^2 0 quad Rightarrow quad x frac{1}{2}]

Hence, there is only one root, ( x frac{1}{2} ), confirming that the equation has two equal roots when ( k 4 ).

Example 2: Conditions for Distinct Real Roots

To determine the value of k for which the quadratic equation ( x^2 - 4x k 0 ) has distinct real roots, we again use the discriminant. For the roots to be distinct and real, the discriminant must be positive:

[D > 0 quad Rightarrow quad (-4)^2 - 4(1)(k) > 0 quad Rightarrow quad 16 - 4k > 0]

Solving for k:

[16 - 4k > 0 quad Rightarrow quad 4k Therefore, for the equation ( x^2 - 4x k 0 ) to have distinct real roots, k must be less than 4.

Generalized Approach

In general, for the quadratic equation ( ax^2 bx c 0 ), the discriminant ( D ) is given by:

[D b^2 - 4ac]

For two equal roots, ( D 0 ), and for distinct real roots, ( D > 0 ).

To summarize, the value of ( k ) that results in the quadratic equation ( kx^2 - 4x 1 0 ) having two equal roots is ( k 4 ). If the equation needs to have distinct real roots, then ( k

Conclusion

This article has explored the concept of the discriminant in quadratic equations and demonstrated how to find the value of ( k ) for specific cases. Understanding the discriminant is essential for determining the nature of the roots of a quadratic equation, making it a critical tool in algebra and mathematical problem-solving.