The given quadratic equation is:

$$(q - r)x^2 + (r - p)x + (p - q) = 0$$

Equation and Coefficients Defined

This is a quadratic equation in the standard form $Ax^2 + Bx + C = 0$, where the coefficients are identified as:

• $$A = (q - r)$$
• $$B = (r - p)$$
• $$C = (p - q)$$

Discriminant for Equal Roots

A fundamental property of quadratic equations is that they have equal roots if and only if their discriminant ($D$) is zero. The formula for the discriminant is:

$$D = B^2 - 4AC$$

Setting the discriminant to zero for equal roots gives:

$$(r - p)^2 - 4(q - r)(p - q) = 0$$

Expanding this expression:

$$(r^2 - 2rp + p^2) - 4(qp - q^2 - rp + rq) = 0$$

$$r^2 - 2rp + p^2 - 4qp + 4q^2 + 4rp - 4rq = 0$$

Combining like terms results in:

$$p^2 + 4q^2 + r^2 + 2rp - 4qp - 4rq = 0$$

While this equation correctly represents the condition, solving it directly can be cumbersome. An alternative method using the sum of coefficients is often more straightforward.

Sum of Coefficients Analysis

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

Sum = $$A + B + C$$

Substitute the coefficient values:

Sum = $$(q - r) + (r - p) + (p - q)$$

Simplify the expression:

Sum = $$q - r + r - p + p - q$$

Group terms:

Sum = $$(q - q) + (r - r) + (p - p)$$

Sum = $$0 + 0 + 0 = 0$$

Root Implication from Sum

A crucial property states that if the sum of the coefficients of a quadratic equation ($Ax^2 + Bx + C = 0$) is zero ($A + B + C = 0$), then $x=1$ is guaranteed to be one of the roots.

We can verify this by substituting $x=1$ into the original equation:

$$(q - r)(1)^2 + (r - p)(1) + (p - q) = (q - r) + (r - p) + (p - q) = 0$$

The problem specifies that the roots of the equation are equal. Since we've established that $x=1$ is a root, and the roots are equal, it follows that both roots must be $1$.

Relation $p + r = 2q$ Derived

For any quadratic equation $Ax^2 + Bx + C = 0$, the product of the roots is given by $C/A$. If both roots are $1$, their product is $1 \times 1 = 1$.

Therefore, we must have:

$$\frac{C}{A} = 1$$

This equality implies that $C = A$. Now, substitute the expressions for $C$ and $A$:

$$p - q = q - r$$

To find the required relationship between $p$, $q$, and $r$, we rearrange this equation:

Add $q$ to both sides:

$$p = q - r + q$$

$$p = 2q - r$$

Add $r$ to both sides:

$$p + r = 2q$$

This condition, $p + r = 2q$, must hold true if the roots of the given quadratic equation are equal. This also indicates that $p$, $q$, and $r$ form an arithmetic progression.