<p>Number of integers satisfying the inequality \(x^4 - 29x^2 + 100 \leq 0\) is</p>
Step-by-Step Solution
Key Concept: Substitute y = x² to convert the quartic inequality into a quadratic, then solve for the range of y and determine which integer values of x satisfy the resulting conditions.
<p><strong>Step 1:</strong> Let y = x². The inequality becomes: y² - 29y + 100 ≤ 0</p><p><strong>Step 2:</strong> Solve y² - 29y + 100 = 0 using the quadratic formula or factoring:<br>y² - 29y + 100 = (y - 4)(y - 25) = 0<br>So y = 4 or y = 25</p><p><strong>Step 3:</strong> Since the coefficient of y² is positive, the parabola opens upward, so:<br>y² - 29y + 100 ≤ 0 when 4 ≤ y ≤ 25</p><p><strong>Step 4:</strong> Substitute back y = x²:<br>4 ≤ x² ≤ 25</p><p><strong>Step 5:</strong> Taking square roots (considering both positive and negative):<br>2 ≤ |x| ≤ 5<br>This means: -5 ≤ x ≤ -2 or 2 ≤ x ≤ 5</p><p><strong>Step 6:</strong> Count integers in these ranges:<br>From [-5, -2]: {-5, -4, -3, -2} = 4 integers<br>From [2, 5]: {2, 3, 4, 5} = 4 integers<br>Total = 8 integers</p><p>∴ Answer: C (8)</p>
Correct Answer: C