Expand (i) `(3x - 2y)^(2)` (ii) `((3)/(4) p - (5)/(6) q)^(2)`

Answer

To expand the given expressions, we will use the algebraic identity for the square of a binomial, which states that:

\[ (a - b)^2 = a^2 - 2ab + b^2 \]

### Solution Steps:

**(i) Expand \((3x - 2y)^2\)**

1. **Identify \(a\) and \(b\)**:
- Here, \(a = 3x\) and \(b = 2y\).

2. **Apply the identity**:
\[ (3x - 2y)^2 = (3x)^2 - 2(3x)(2y) + (2y)^2 \]

3. **Calculate \(a^2\)**:
\[ (3x)^2 = 9x^2 \]

4. **Calculate \(b^2\)**:
\[ (2y)^2 = 4y^2 \]

5. **Calculate \(2ab\)**:
\[ 2(3x)(2y) = 12xy \]

6. **Combine the results**:
\[ (3x - 2y)^2 = 9x^2 - 12xy + 4y^2 \]

**Final Answer for (i)**:
\[ (3x - 2y)^2 = 9x^2 - 12xy + 4y^2 \]

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**(ii) Expand \(\left(\frac{3}{4}p - \frac{5}{6}q\right)^2\)**

1. **Identify \(a\) and \(b\)**:
- Here, \(a = \frac{3}{4}p\) and \(b = \frac{5}{6}q\).

2. **Apply the identity**:
\[ \left(\frac{3}{4}p - \frac{5}{6}q\right)^2 = \left(\frac{3}{4}p\right)^2 - 2\left(\frac{3}{4}p\right)\left(\frac{5}{6}q\right) + \left(\frac{5}{6}q\right)^2 \]

3. **Calculate \(a^2\)**:
\[ \left(\frac{3}{4}p\right)^2 = \frac{9}{16}p^2 \]

4. **Calculate \(b^2\)**:
\[ \left(\frac{5}{6}q\right)^2 = \frac{25}{36}q^2 \]

5. **Calculate \(2ab\)**:
\[ 2\left(\frac{3}{4}p\right)\left(\frac{5}{6}q\right) = \frac{30}{24}pq = \frac{5}{4}pq \]

6. **Combine the results**:
\[ \left(\frac{3}{4}p - \frac{5}{6}q\right)^2 = \frac{9}{16}p^2 - \frac{5}{4}pq + \frac{25}{36}q^2 \]

**Final Answer for (ii)**:
\[ \left(\frac{3}{4}p - \frac{5}{6}q\right)^2 = \frac{9}{16}p^2 - \frac{5}{4}pq + \frac{25}{36}q^2 \]

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