https://file.aiquickdraw.com/mathbot/file/1730075304272udifpfpj.jpg

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.

Solve the following quadratic inequality: 4x^2 - 12x + 9 <= 0. Write your answer in interval notation.

Learn how to solve the quadratic inequality 4x^2 - 12x + 9 <= 0. This step-by-step guide covers factoring techniques and interval notation, helping students understand the solution of quadratic inequalities. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation