To solve the inequality $4x^2 - 12x + 9 \leq 0$, we can approach it by solving the associated quadratic equation and then analyzing the intervals.

Step-by-Step Solution

1. Set the Inequality to an Equation: $4x^2 - 12x + 9 = 0$

2. Solve the Quadratic Equation: This quadratic equation can be factored as follows: $(2x - 3)^2 = 0$ Thus, $2x - 3 = 0$ which gives $x = \frac{3}{2}$.

   So, the quadratic equation has a repeated root at $x = \frac{3}{2}$.

3. Analyze the Sign of the Quadratic Expression: Since $(2x - 3)^2$ is a perfect square, it is always non-negative (i.e., $\geq 0$). Therefore, the expression $4x^2 - 12x + 9$ will be zero only at $x = \frac{3}{2}$ and positive for all other $x$ values.

4. Apply the Inequality: We are looking for where $4x^2 - 12x + 9 \leq 0$, which occurs only when: $x = \frac{3}{2}$

Conclusion

The solution in interval notation is: $\left\{ \frac{3}{2} \right\}$ or just $\frac{3}{2}$ if only the point notation is accepted.

Would you like further details or have any questions?

Related Questions

1. How do you determine if a quadratic expression is always positive or negative?
2. What are the properties of perfect square trinomials and their effect on inequalities?
3. Can a quadratic inequality have more than one interval as a solution?
4. How would the solution change if the inequality were strict (i.e., $<$ instead of $\leq$)?
5. What happens if the quadratic has two distinct real roots?

Tip

For quadratic inequalities, always check if the quadratic expression can be factored as a perfect square. This simplifies the analysis significantly.