4х²-8х≤0

To solve the quadratic inequality \( 4x^2 - 8x \leq 0 \), first, we can factor the left-hand side.

1. Factor the expression:

   \[ 4x^2 - 8x = 4x(x - 2) \]

   So, the inequality becomes:

   \[ 4x(x - 2) \leq 0 \]

2. Find the zeros of the expression:

   Set \( 4x(x - 2) = 0 \):

   This gives us:

   \[ x = 0 \quad \text{or} \quad x - 2 = 0 \implies x = 2 \]

   So, the critical points are \( x = 0 \) and \( x = 2 \).

3. Test intervals:

   We will test the intervals determined by the critical points \( (-\infty, 0) \), \( (0, 2) \), and \( (2, \infty) \).

   • Interval \( (-\infty, 0) \): Choose \( x = -1 \), \[ 4(-1)(-1 - 2) = 4(-1)(-3) = 12 > 0 \]

   • Interval \( (0, 2) \): Choose \( x = 1 \), \[ 4(1)(1 - 2) = 4(1)(-1) = -4 \leq 0 \]

   • Interval \( (2, \infty) \): Choose \( x = 3 \), \[ 4(3)(3 - 2) = 4(3)(1) = 12 > 0 \]

4. Determine the solution set:

   From the tests, we find that \( 4x(x - 2) \leq 0 \) in the interval \( [0, 2] \).

   Therefore, the solution to the inequality \( 4x^2 - 8x \leq 0 \) is:

   \[ x \in [0, 2] \]