Number of intergers satisfying the inequality
`x^4- 29x^2+100 le 0 `is

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To solve the inequality \( x^4 - 29x^2 + 100 \leq 0 \), we will follow these steps:

### Step 1: Substitute \( x^2 \) with \( t \)

Let \( t = x^2 \). Then the inequality becomes:
\[ t^2 - 29t + 100 \leq 0 \]

### Step 2: Factor the quadratic expression

We need to factor the quadratic \( t^2 - 29t + 100 \). We can rewrite it as:
\[ t^2 - 25t - 4t + 100 \leq 0 \]
Now, we can factor by grouping:
\[ t(t - 25) - 4(t - 25) \leq 0 \]
This gives us:
\[ (t - 4)(t - 25) \leq 0 \]

### Step 3: Find the critical points

The critical points from the factors are \( t = 4 \) and \( t = 25 \).

### Step 4: Determine the intervals

We need to analyze the sign of the expression \( (t - 4)(t - 25) \) in the intervals determined by the critical points:
- \( (-\infty, 4) \)
- \( [4, 25] \)
- \( (25, \infty) \)

### Step 5: Test the intervals

1. For \( t < 4 \) (e.g., \( t = 0 \)):
\[ (0 - 4)(0 - 25) = 100 > 0 \]
2. For \( 4 \leq t \leq 25 \) (e.g., \( t = 10 \)):
\[ (10 - 4)(10 - 25) = 6 \cdot (-15) = -90 < 0 \]
3. For \( t > 25 \) (e.g., \( t = 30 \)):
\[ (30 - 4)(30 - 25) = 26 \cdot 5 = 130 > 0 \]

### Step 6: Write the solution for \( t \)

The solution for the inequality \( (t - 4)(t - 25) \leq 0 \) is:
\[ 4 \leq t \leq 25 \]

### Step 7: Substitute back for \( x \)

Since \( t = x^2 \), we have:
\[ 4 \leq x^2 \leq 25 \]

### Step 8: Solve for \( x \)

This translates to:
\[ x^2 \geq 4 \quad \text{and} \quad x^2 \leq 25 \]
Taking square roots, we get:
\[ x \geq 2 \quad \text{or} \quad x \leq -2 \]
And:
\[ -5 \leq x \leq 5 \]

### Step 9: Find the intersection of the intervals

Now we combine these conditions:
1. \( x \geq 2 \) or \( x \leq -2 \)
2. \( -5 \leq x \leq 5 \)

The valid intervals are:
- From \( -5 \) to \( -2 \) (inclusive)
- From \( 2 \) to \( 5 \) (inclusive)

### Step 10: Count the integers

The integers in the intervals:
- From \( -5 \) to \( -2 \): \( -5, -4, -3, -2 \) (4 integers)
- From \( 2 \) to \( 5 \): \( 2, 3, 4, 5 \) (4 integers)

Thus, the total number of integers satisfying the inequality is:
\[ 4 + 4 = 8 \]

### Final Answer:
The number of integers satisfying the inequality \( x^4 - 29x^2 + 100 \leq 0 \) is **8**.