If `x^(2) , x, -8` are in AP, then which one of the following is correct?

Options

A. `x in {-2}`
B. `x in {4}`
C. `x in {-2, 4}`
D. `x in {-4, 2}`

Answer

To determine the values of \( x \) for which \( x^2, x, -8 \) are in Arithmetic Progression (AP), we follow these steps:

### Step 1: Understand the condition for AP
For three numbers \( a, b, c \) to be in AP, the condition is:
\[ 2b = a + c \]
In our case, \( a = x^2 \), \( b = x \), and \( c = -8 \).

### Step 2: Set up the equation
Using the AP condition, we substitute the values:
\[ 2x = x^2 + (-8) \]
This simplifies to:
\[ 2x = x^2 - 8 \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ x^2 - 2x - 8 = 0 \]

### Step 4: Apply the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -2 \), and \( c = -8 \).

### Step 5: Calculate the discriminant
First, we calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot (-8) = 4 + 32 = 36 \]

### Step 6: Find the roots
Now substituting into the quadratic formula:
\[ x = \frac{-(-2) \pm \sqrt{36}}{2 \cdot 1} = \frac{2 \pm 6}{2} \]
This gives us two possible solutions:
1. \( x = \frac{2 + 6}{2} = \frac{8}{2} = 4 \)
2. \( x = \frac{2 - 6}{2} = \frac{-4}{2} = -2 \)

### Step 7: Conclusion
Thus, the values of \( x \) that satisfy the condition are:
\[ x = 4 \quad \text{and} \quad x = -2 \]