Understanding the Quadratic Equation with Equal Roots

We are given a quadratic equation \(a(b - c) x^2 + b (c - a) x + c (a - b) = 0\). The question states that the roots of this equation are equal. We need to find the relationship between the coefficients \(a\), \(b\), and \(c\).

For a quadratic equation to have equal roots, its discriminant must be zero. However, let's first analyze the structure of this specific equation.

Analyzing the Coefficients of the Given Equation

Let the given quadratic equation be in the standard form \(Ax^2 + Bx + C = 0\). By comparing the given equation with the standard form, we can identify the coefficients:

• The coefficient of \(x^2\) is \(A = a(b - c)\).
• The coefficient of \(x\) is \(B = b (c - a)\).
• The constant term is \(C = c (a - b)\).

The Significance of the Sum of Coefficients

Let's calculate the sum of the coefficients \(A\), \(B\), and \(C\):

Expanding the terms, we get:

Now, let's rearrange and cancel the terms:

A key property of quadratic equations is that if the sum of its coefficients \(A + B + C\) is equal to zero, then \(x = 1\) is one of the roots of the equation \(Ax^2 + Bx + C = 0\).

Properties of Equal Roots in Quadratic Equations

The problem explicitly states that the roots of the equation are equal. We have just discovered that one root of the equation is \(x = 1\). Since the roots are equal, this means both roots must have the same value. Therefore, both roots of the given quadratic equation must be \(x = 1\).

For a quadratic equation \(Ax^2 + Bx + C = 0\) with roots \(\alpha\) and \(\beta\), the following relationships hold:

• Sum of roots: \(\alpha + \beta = \frac{-B}{A}\)
• Product of roots: \(\alpha \beta = \frac{C}{A}\)

Since both roots are 1, we have \(\alpha = 1\) and \(\beta = 1\).

Deriving the Relationship for a, b, c in HP

Let's use the property of the product of roots:

So, we must have:

Now, substitute the expressions for C and A in terms of \(a\), \(b\), and \(c\):

Expand both sides:

Rearrange the terms to find the relationship between \(a\), \(b\), and \(c\). Let's move the \(-ac\) from the right side to the left and \(-bc\) from the left side to the right:

This equation, \(2ac = ab + bc\), is the condition we were looking for. To understand what this means for the sequence \(a, b, c\), we can divide the entire equation by \(abc\) (assuming \(a, b, c\) are non-zero, which must be true if the equation is a valid quadratic with equal roots where \(a(b-c) \neq 0\)).

Simplifying each term:

We can rewrite this as:

This equation shows that \(\frac{1}{b}\) is the arithmetic mean of \(\frac{1}{a}\) and \(\frac{1}{c}\). This is the defining condition for three numbers \(a\), \(b\), and \(c\) to be in Harmonic Progression (HP). Three numbers are in HP if their reciprocals form an Arithmetic Progression (AP).

We could also use the sum of roots property \(\frac{-B}{A} = 2 \implies B = -2A\). Substituting the expressions: \(b(c-a) = -2a(b-c) \implies bc - ab = -2ab + 2ac \implies bc + ab - 2ac = 0\), which is the same condition \(2ac = ab + bc\).

Conclusion: Equal Roots and Harmonic Progression

Thus, if the roots of the equation \(a(b - c) x^2 + b (c - a) x + c (a - b) = 0\) are equal, the relationship \(\frac{1}{a} + \frac{1}{c} = \frac{2}{b}\) holds, which implies that \(a\), \(b\), and \(c\) are in Harmonic Progression (HP).

Revision Table: Quadratic Roots and Progressions

Additional Information: Understanding AP, GP, and HP

Arithmetic, Geometric, and Harmonic Progressions are types of sequences where terms follow specific rules.

• Arithmetic Progression (AP): A sequence where each term after the first is obtained by adding a fixed constant, called the common difference, to the preceding term. If \(a, b, c\) are in AP, then \(b-a = c-b\), which means \(2b = a+c\). \(b\) is the arithmetic mean of \(a\) and \(c\).
• Geometric Progression (GP): A sequence of non-zero numbers where each term after the first is obtained by multiplying the preceding term by a fixed non-zero constant, called the common ratio. If \(a, b, c\) are in GP, then \(\frac{b}{a} = \frac{c}{b}\), which means \(b^2 = ac\). \(b\) is the geometric mean of \(a\) and \(c\).
• Harmonic Progression (HP): A sequence of non-zero numbers whose reciprocals form an arithmetic progression. If \(a, b, c\) are in HP, then \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) are in AP. This implies \(\frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}\), leading to \(\frac{2}{b} = \frac{1}{a} + \frac{1}{c}\). \(\frac{1}{b}\) is the arithmetic mean of \(\frac{1}{a}\) and \(\frac{1}{c}\).

Our analysis of the given quadratic equation with equal roots led directly to the condition \(\frac{2}{b} = \frac{1}{a} + \frac{1}{c}\), confirming that \(a, b, c\) are in Harmonic Progression.