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