If `b gt a`, then the equation `(x-a)(x-b) - 1 = 0` has,

Options

A. both roots in [a, b]
B. both roots in `(-oo, 0)`
C. both roots in `(b, +oo)`
D. one root in `(-oo, a)` and other root in `(b, +oo)`

Answer

To solve the equation \((x-a)(x-b) - 1 = 0\) given that \(b > a\), we will follow these steps:

### Step 1: Expand the Equation
We start by expanding the left-hand side of the equation:

\[ (x-a)(x-b) - 1 = 0 \]

Expanding \((x-a)(x-b)\):

\[ x^2 - (a+b)x + ab - 1 = 0 \]

### Step 2: Rewrite the Equation
Now, we can rewrite the equation in standard quadratic form:

\[ x^2 - (a+b)x + (ab - 1) = 0 \]

### Step 3: Identify Coefficients
In the quadratic equation \(Ax^2 + Bx + C = 0\), we identify:

- \(A = 1\)
- \(B = -(a+b)\)
- \(C = ab - 1\)

### Step 4: Calculate the Discriminant
The discriminant \(D\) of a quadratic equation is given by:

\[ D = B^2 - 4AC \]

Substituting the values of \(A\), \(B\), and \(C\):

\[ D = (-(a+b))^2 - 4(1)(ab - 1) \]

This simplifies to:

\[ D = (a+b)^2 - 4(ab - 1) \]

### Step 5: Simplify the Discriminant
Now, we simplify the expression for the discriminant:

\[ D = (a^2 + 2ab + b^2) - (4ab - 4) \]
\[ D = a^2 + 2ab + b^2 - 4ab + 4 \]
\[ D = a^2 - 2ab + b^2 + 4 \]
\[ D = (a - b)^2 + 4 \]

### Step 6: Analyze the Discriminant
Since \((a - b)^2 \geq 0\), it follows that:

\[ D \geq 4 \]

This means the discriminant is always positive, indicating that the quadratic equation has two distinct real roots.

### Conclusion
Thus, the equation \((x-a)(x-b) - 1 = 0\) has **two distinct real roots** when \(b > a\).

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