If c and d are roots of the equation (x-a) (x-b) - k = 0, then a, b are roots of the equation

Options

A. `(x-c) (x-d) - k = 0`
B. `(x-c) (x-d) + k = 0`
C. `(x-a) (x-c) + k = 0`
D. `(x-b) (x-d) + k = 0`

Answer

To solve the problem step by step, we need to analyze the given quadratic equation and derive the new equation whose roots are \( a \) and \( b \).

### Step 1: Understand the given equation
The given equation is:
\[ (x - a)(x - b) - k = 0 \]
This can be expanded to:
\[ x^2 - (a + b)x + (ab - k) = 0 \]
From this equation, we can identify the sum and product of the roots.

### Step 2: Identify the roots
Let \( c \) and \( d \) be the roots of the equation. According to Vieta's formulas:
- The sum of the roots \( c + d = a + b \)
- The product of the roots \( cd = ab - k \)

### Step 3: Form the new equation
We need to find the equation for which \( a \) and \( b \) are the roots. Using Vieta's formulas again, we can express the new equation in terms of its roots:
- The sum of the roots \( a + b = c + d \)
- The product of the roots \( ab = cd + k \)

### Step 4: Write the new quadratic equation
Using the sum and product of the roots, we can write the new quadratic equation as:
\[ x^2 - (a + b)x + ab = 0 \]
Substituting the values we found:
\[ x^2 - (c + d)x + (cd + k) = 0 \]

### Final Form of the Equation
Thus, the equation for which \( a \) and \( b \) are roots is:
\[ x^2 - (c + d)x + (cd + k) = 0 \]