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The given expression is:

\[
4x^2 - 6x + \frac{1}{4}
\]

The question asks which number should be added to this expression to form a perfect square trinomial.

A perfect square trinomial has the form \( a(x + b)^2 \), where we need to complete the square. Let's break down the expression:

1. Start with the quadratic part: \( 4x^2 - 6x \).
2. Factor out the leading coefficient \( 4 \) from the first two terms:

   \[
   4(x^2 - \frac{3}{2}x) + \frac{1}{4}
   \]

3. To complete the square, take half of the coefficient of \( x \), square it, and add it inside the parenthesis. Half of \( \frac{3}{2} \) is \( \frac{3}{4} \), and squaring it gives \( \left(\frac{3}{4}\right)^2 = \frac{9}{16} \).

4. Add and subtract \( \frac{9}{16} \) inside the parenthesis:

   \[
   4\left(x^2 - \frac{3}{2}x + \frac{9}{16} - \frac{9}{16}\right) + \frac{1}{4}
   \]

5. Simplify:

   \[
   4\left(\left(x - \frac{3}{4}\right)^2 - \frac{9}{16}\right) + \frac{1}{4}
   \]

6. Distribute the \( 4 \):

   \[
   4\left(x - \frac{3}{4}\right)^2 - \frac{36}{16} + \frac{1}{4}
   \]

7. Combine the constants:

   \[
   4\left(x - \frac{3}{4}\right)^2 - \frac{36}{16} + \frac{4}{16} = 4\left(x - \frac{3}{4}\right)^2 - \frac{32}{16} = 4\left(x - \frac{3}{4}\right)^2 - 2
   \]

To make this a perfect square trinomial, we need to add 2 to the expression to cancel out the constant term \( -2 \).

### Conclusion:
The number that should be added to the expression to form a perfect square trinomial is **2**.

Let me know if you would like more details or have any questions. Here are 5 related questions to deepen your understanding:

1. How do you factor a perfect square trinomial in general?
2. Can you explain the process of completing the square step by step?
3. What is the role of adding and subtracting a constant when completing the square?
4. How does the constant inside the parentheses affect the completed square?
5. How do you solve quadratic equations by completing the square?

**Tip:** Completing the square is particularly useful in solving quadratic equations and deriving the quadratic formula. Practice this method to improve your problem-solving skills!

با افزودن کدام عدد به عبارت 4x^2 - 6x + 1/4 مربع یک دوجمله‌ای حاصل می‌شود؟

Learn how to solve the quadratic expression 4x^2 - 6x + 1/4 by completing the square. This algebra problem helps you understand the process of transforming a quadratic equation into a perfect square trinomial. Suitable for high school students in Grades 9-12.

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