Minimum possible number of positive root of the quadratic equation `x^2 -(1 +lambda)x+ lambda -2 =0 ` , `lambda in R`

Options

A. 2
B. 0
C. 1
D. can not be determined

Answer

To solve the problem of finding the minimum possible number of positive roots of the quadratic equation

\[ x^2 - (1 + \lambda)x + (\lambda - 2) = 0 \]

where \(\lambda \in \mathbb{R}\), we will follow these steps:

### Step 1: Identify the coefficients of the quadratic equation
The given quadratic equation can be expressed in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = -(1 + \lambda) \)
- \( c = \lambda - 2 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) of a quadratic equation is given by the formula:

\[ D = b^2 - 4ac \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = (-(1 + \lambda))^2 - 4(1)(\lambda - 2) \]

\[ D = (1 + \lambda)^2 - 4(\lambda - 2) \]

Expanding this:

\[ D = 1 + 2\lambda + \lambda^2 - 4\lambda + 8 \]

\[ D = \lambda^2 - 2\lambda + 9 \]

### Step 3: Analyze the discriminant
The discriminant \( D = \lambda^2 - 2\lambda + 9 \) is a quadratic expression. To determine its nature, we can complete the square:

\[ D = (\lambda - 1)^2 + 8 \]

Since \((\lambda - 1)^2 \geq 0\), it follows that \(D \geq 8\). Therefore, the discriminant is always positive, indicating that the quadratic equation has two real roots for any value of \(\lambda\).

### Step 4: Determine conditions for positive roots
Next, we need to find conditions under which the roots are positive. For a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( S = -\frac{b}{a} \)
- The product of the roots \( P = \frac{c}{a} \)

Calculating the sum and product of the roots:

\[ S = -\frac{-(1 + \lambda)}{1} = 1 + \lambda \]
\[ P = \frac{\lambda - 2}{1} = \lambda - 2 \]

### Step 5: Analyze cases for positive roots
1. **Case 1: No positive roots**
- For both roots to be negative, we require:
- \( S < 0 \) (sum of roots is negative)
- \( P > 0 \) (product of roots is positive)
- From \( S < 0 \):
\[ 1 + \lambda < 0 \implies \lambda < -1 \]
- From \( P > 0 \):
\[ \lambda - 2 > 0 \implies \lambda > 2 \]
- There are no values of \(\lambda\) that satisfy both conditions simultaneously. Thus, it is impossible for both roots to be negative.

2. **Case 2: One positive root**
- For one root to be positive and the other negative, we require:
- \( S > 0 \)
- \( P < 0 \)
- From \( S > 0 \):
\[ 1 + \lambda > 0 \implies \lambda > -1 \]
- From \( P < 0 \):
\[ \lambda - 2 < 0 \implies \lambda < 2 \]
- The values of \(\lambda\) that satisfy both conditions are:
\[ -1 < \lambda < 2 \]
- Therefore, it is possible to have one positive root in this range.

3. **Case 3: Two positive roots**
- For both roots to be positive, we require:
- \( S > 0 \)
- \( P > 0 \)
- From \( S > 0 \):
\[ 1 + \lambda > 0 \implies \lambda > -1 \]
- From \( P > 0 \):
\[ \lambda - 2 > 0 \implies \lambda > 2 \]
- The values of \(\lambda\) that satisfy both conditions are:
\[ \lambda > 2 \]
- Therefore, it is possible to have two positive roots when \(\lambda > 2\).

### Conclusion
The minimum possible number of positive roots of the given quadratic equation is **1**.