To express \( 9x^2 - 15x \) in the form \((3x - a)^2 - b\), we follow these steps:

1. **Rewrite the expression in a form that resembles the expansion of a binomial square.**
2. **Identify the values for \( a \) and \( b \) to complete the square.**

The expression \( (3x - a)^2 \) expands to \( 9x^2 - 6ax + a^2 \). We need our given expression \( 9x^2 - 15x \) to fit this format. 

### Step-by-Step Solution:

1. **Start with the given expression:**

   \[ 9x^2 - 15x \]

2. **Factor out the common factor of the quadratic coefficient (9):**

   \[ 9x^2 - 15x = 9(x^2 - \frac{15}{9}x) = 9(x^2 - \frac{5}{3}x) \]

3. **Complete the square inside the parentheses:**

   To complete the square for \( x^2 - \frac{5}{3}x \), we take half of the coefficient of \( x \), square it, and add and subtract it inside the parentheses:

   \[ x^2 - \frac{5}{3}x \]
   \[ = x^2 - \frac{5}{3}x + \left(\frac{-\frac{5}{3}}{2}\right)^2 - \left(\frac{-\frac{5}{3}}{2}\right)^2 \]
   \[ = x^2 - \frac{5}{3}x + \left(\frac{-5}{6}\right)^2 - \left(\frac{-5}{6}\right)^2 \]
   \[ = x^2 - \frac{5}{3}x + \frac{25}{36} - \frac{25}{36} \]
   \[ = \left(x - \frac{5}{6}\right)^2 - \frac{25}{36} \]

4. **Multiply through by 9:**

   \[ 9\left(\left(x - \frac{5}{6}\right)^2 - \frac{25}{36}\right) = 9\left(x - \frac{5}{6}\right)^2 - 9 \cdot \frac{25}{36} \]
   \[ = 9\left(x - \frac{5}{6}\right)^2 - \frac{225}{36} \]
   \[ = 9\left(x - \frac{5}{6}\right)^2 - \frac{25}{4} \]

5. **Rewrite the expression in the desired form:**

   \[ 9x^2 - 15x = \left(3x - \frac{15}{6}\right)^2 - \frac{25}{4} \]
   \[ = (3x - \frac{5}{2})^2 - \frac{25}{4} \]

So, the expression \( 9x^2 - 15x \) in the form \((3x - a)^2 - b\) is:

\[ 9x^2 - 15x = (3x - \frac{5}{2})^2 - \frac{25}{4} \]

Would you like further details or have any questions?

Here are 8 related questions to expand your understanding:

1. How do you complete the square for a general quadratic expression?
2. Can you express \( 4x^2 - 12x + 7 \) in a similar completed square form?
3. What is the significance of completing the square in solving quadratic equations?
4. How can you use the method of completing the square to find the vertex of a parabola?
5. Can you derive the quadratic formula using the method of completing the square?
6. What is the geometric interpretation of the form \((3x - a)^2 - b\)?
7. How do you solve \( 9x^2 - 15x = 0 \) using the completed square form?
8. What are some real-life applications of completing the square?

**Tip:** When completing the square, always ensure to balance the equation by adding and subtracting the same value inside the parentheses.

Learn how to express 9x^2 - 15x in the form (3x - 5/2)^2 - 25/4 using the method of completing the square. This problem covers key algebraic concepts, including completing the square, quadratic equations, and factorization. Suitable for students in Grades 10-12.

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