We are given the quadratic equation:

$4x^2 - 12x + 9 = 0$

Step 1: Recognize that this is a perfect square

Notice that this equation can be factored as a perfect square. Let's rewrite it:

$(2x - 3)^2 = 0$

You can check this by expanding $(2x - 3)^2$:

$(2x - 3)(2x - 3) = 4x^2 - 6x - 6x + 9 = 4x^2 - 12x + 9$

This shows that the equation is a perfect square.

Step 2: Solve the equation

Since $(2x - 3)^2 = 0$, we can take the square root of both sides:

$2x - 3 = 0$

Step 3: Solve for $x$

Add 3 to both sides:

$2x = 3$

Now, divide both sides by 2:

$x = \frac{3}{2}$

Final Answer:

The solution to the equation is:

$x = \frac{3}{2}$

Recap:

1. Recognize the quadratic as a perfect square.
2. Factor the quadratic expression.
3. Solve by setting the factor equal to zero and solving for $x$.

Would you like more details or additional practice?

Here are 5 related questions for further practice:

1. How do you solve $x^2 - 6x + 9 = 0$?
2. Can you factor and solve $9x^2 - 12x + 4 = 0$?
3. How do you solve a quadratic equation like $x^2 + 4x - 12 = 0$?
4. What happens when the quadratic does not factor easily, like $2x^2 - 3x + 5 = 0$?
5. How do you solve quadratic equations using the quadratic formula?

Tip: Always check if a quadratic equation can be factored as a perfect square. This can save time and simplify the solving process.