Question

Verify that roots of the quadratic equation (p – q)x\(^2\) + (q – r)x + (r – p) = 0 are equal when q + r = 2p.

Hint:

An alternative way to solve this is to substitute \(p = \frac{q + r}{2}\) directly into the discriminant formula \(D = B^2 - 4AC\) and show that it reduces to zero.
However, utilizing the root properties of equations with coefficient sums equal to zero is much faster and reduces algebra steps!

Answer 1

Step 1: Understanding the Question:
The topic of this question is Quadratic Equations.
A quadratic equation of the form \(Ax^2 + Bx + C = 0\) has equal roots if and only if its discriminant \(D = B^2 - 4AC\) is equal to zero.
We are given the equation \((p - q)x^2 + (q - r)x + (r - p) = 0\) and need to verify that its roots are equal under the condition \(q + r = 2p\).

Step 2: Key Formula or Approach:
Identify the coefficients from the given equation:
\(A = p - q\)
\(B = q - r\)
\(C = r - p\)
Notice that the sum of the coefficients is:
\[ A + B + C = (p - q) + (q - r) + (r - p) = 0 \] Whenever the sum of the coefficients of a quadratic equation is zero, \(x = 1\) is always one of its roots.
For the roots to be equal, the second root must also be equal to 1.
We will use the product of roots formula to verify the given condition:
\[ \text{Product of roots} = \frac{C}{A} \]

Step 3: Detailed Explanation:

• Since the sum of the coefficients is exactly zero:
\[ (p - q) + (q - r) + (r - p) = 0 \] This implies that one root of the quadratic equation is \(x_1 = 1\).

• Since we are given that the roots of the quadratic equation are equal:
The second root must also be \(x_2 = 1\).

• Write down the formula for the product of the roots of a quadratic equation:
\[ x_1 \times x_2 = \frac{C}{A} \] \[ x_1 \times x_2 = \frac{r - p}{p - q} \]

• Substitute \(x_1 = 1\) and \(x_2 = 1\) into the equation:
\[ 1 \times 1 = \frac{r - p}{p - q} \] \[ 1 = \frac{r - p}{p - q} \]

• Cross-multiply to simplify the equation:
\[ p - q = r - p \]

• Group the terms to isolate \(2p\):
\[ p + p = q + r \] \[ q + r = 2p \] This verifies the given condition.


Step 4: Final Answer:
Hence, it is verified that the roots of the quadratic equation are equal when \(q + r = 2p\).