If x is ral and `4y ^(2)+4 xy +x +6=0,` then the complete set of values of x for which y is real, is :

Options

A. `x le - 2 or x ge 3`
B. `x le 2 or x ge 3`
C. `x le - 3 or x ge2`
D. `-3 le x le 2`

Answer

To solve the problem, we need to determine the values of \( x \) for which \( y \) is real in the quadratic equation:

\[ 4y^2 + 4xy + x + 6 = 0 \]

### Step 1: Identify the coefficients of the quadratic equation
The given equation can be viewed as a quadratic in \( y \):
\[ a = 4, \quad b = 4x, \quad c = x + 6 \]

### Step 2: Write the condition for \( y \) to be real
For \( y \) to be real, the discriminant of the quadratic equation must be greater than or equal to zero:
\[ D = b^2 - 4ac \geq 0 \]

### Step 3: Substitute the coefficients into the discriminant formula
Substituting \( a \), \( b \), and \( c \) into the discriminant:
\[ D = (4x)^2 - 4(4)(x + 6) \geq 0 \]

### Step 4: Simplify the discriminant
Calculating \( D \):
\[ D = 16x^2 - 16(x + 6) \geq 0 \]
\[ D = 16x^2 - 16x - 96 \geq 0 \]

### Step 5: Factor out the common term
Factoring out 16:
\[ 16(x^2 - x - 6) \geq 0 \]
Dividing both sides by 16 (since 16 is positive, the inequality remains the same):
\[ x^2 - x - 6 \geq 0 \]

### Step 6: Factor the quadratic expression
Now, we factor the quadratic:
\[ (x + 2)(x - 3) \geq 0 \]

### Step 7: Find the critical points
The critical points are \( x = -2 \) and \( x = 3 \).

### Step 8: Determine the intervals
We need to test the intervals determined by the critical points:
1. \( (-\infty, -2) \)
2. \( (-2, 3) \)
3. \( (3, \infty) \)

### Step 9: Test each interval
- For \( x < -2 \) (e.g., \( x = -3 \)):
\[ (-3 + 2)(-3 - 3) = (-1)(-6) > 0 \]
- For \( -2 < x < 3 \) (e.g., \( x = 0 \)):
\[ (0 + 2)(0 - 3) = (2)(-3) < 0 \]
- For \( x > 3 \) (e.g., \( x = 4 \)):
\[ (4 + 2)(4 - 3) = (6)(1) > 0 \]

### Step 10: Write the solution
The solution to the inequality \( (x + 2)(x - 3) \geq 0 \) is:
\[ x \in (-\infty, -2] \cup [3, \infty) \]

### Final Answer
Thus, the complete set of values of \( x \) for which \( y \) is real is:
\[ x \leq -2 \quad \text{or} \quad x \geq 3 \]

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