The equation `(x^(2)-6x+8)+lambda (x^(2)-4x+3)=0, lambda in R` has

Options

A. real and unequal roots for all `lambda`
B. real roots for `lambda lt 0`
C. real roots for `lambda gt 0`
D. real and unequal roots for `lambda = 0`

Answer

To solve the equation \( (x^2 - 6x + 8) + \lambda (x^2 - 4x + 3) = 0 \) where \( \lambda \in \mathbb{R} \), we will follow these steps:

### Step 1: Combine the equations
We start by combining the two quadratic expressions:

\[ (1 + \lambda)x^2 + (-6 - 4\lambda)x + (8 + 3\lambda) = 0 \]

Here, the coefficients are:
- \( a = 1 + \lambda \)
- \( b = -6 - 4\lambda \)
- \( c = 8 + 3\lambda \)

### Step 2: Calculate the Discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

Substituting our coefficients into the discriminant formula:

\[ D = (-6 - 4\lambda)^2 - 4(1 + \lambda)(8 + 3\lambda) \]

### Step 3: Expand the Discriminant
Now, we will expand the discriminant:

1. Calculate \( (-6 - 4\lambda)^2 \):

\[ (-6 - 4\lambda)^2 = 36 + 48\lambda + 16\lambda^2 \]

2. Calculate \( 4(1 + \lambda)(8 + 3\lambda) \):

\[ 4(1 + \lambda)(8 + 3\lambda) = 4(8 + 3\lambda + 8\lambda + 3\lambda^2) = 32 + 44\lambda + 12\lambda^2 \]

3. Combine these results:

\[ D = (36 + 48\lambda + 16\lambda^2) - (32 + 44\lambda + 12\lambda^2) \]

### Step 4: Simplify the Discriminant
Now we simplify:

\[ D = 36 - 32 + 48\lambda - 44\lambda + 16\lambda^2 - 12\lambda^2 \]
\[ D = 4 + 4\lambda + 4\lambda^2 \]

### Step 5: Factor the Discriminant
We can factor out the common term:

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

### Step 6: Analyze the Discriminant
To determine the nature of the roots, we need to analyze \( \lambda^2 + \lambda + 1 \):

The discriminant of this quadratic is:

\[ b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]

Since the discriminant is negative, \( \lambda^2 + \lambda + 1 \) is always positive for all \( \lambda \in \mathbb{R} \).

### Step 7: Conclusion on Roots
Since \( D = 4(\lambda^2 + \lambda + 1) > 0 \) for all \( \lambda \), it implies that the quadratic equation has:

- **Real and distinct roots for all values of \( \lambda \)**.