In each of the following find the value of 'a' :
`(i) 4x^(2)+ax+9=(2x+3)^(2)`
`(ii) 9x^(2)+(7a-5)x+25=(3x+5)^(2)`

Answer

Let's solve the given question step by step.

### Part (i)
We need to find the value of 'a' in the equation:
\[ 4x^2 + ax + 9 = (2x + 3)^2 \]

**Step 1: Expand the right-hand side**
Using the identity \((a + b)^2 = a^2 + 2ab + b^2\):
\[ (2x + 3)^2 = (2x)^2 + 2(2x)(3) + (3)^2 \]
Calculating each term:
\[ (2x)^2 = 4x^2, \quad 2(2x)(3) = 12x, \quad (3)^2 = 9 \]
So,
\[ (2x + 3)^2 = 4x^2 + 12x + 9 \]

**Step 2: Set the expanded form equal to the left-hand side**
Now we have:
\[ 4x^2 + ax + 9 = 4x^2 + 12x + 9 \]

**Step 3: Compare coefficients**
Since the \(4x^2\) and \(9\) terms are the same on both sides, we can focus on the coefficients of \(x\):
\[ ax = 12x \]
This implies:
\[ a = 12 \]

### Part (ii)
Now we need to find the value of 'a' in the equation:
\[ 9x^2 + (7a - 5)x + 25 = (3x + 5)^2 \]

**Step 1: Expand the right-hand side**
Using the same identity:
\[ (3x + 5)^2 = (3x)^2 + 2(3x)(5) + (5)^2 \]
Calculating each term:
\[ (3x)^2 = 9x^2, \quad 2(3x)(5) = 30x, \quad (5)^2 = 25 \]
So,
\[ (3x + 5)^2 = 9x^2 + 30x + 25 \]

**Step 2: Set the expanded form equal to the left-hand side**
Now we have:
\[ 9x^2 + (7a - 5)x + 25 = 9x^2 + 30x + 25 \]

**Step 3: Compare coefficients**
Since the \(9x^2\) and \(25\) terms are the same on both sides, we can focus on the coefficients of \(x\):
\[ (7a - 5)x = 30x \]
This implies:
\[ 7a - 5 = 30 \]

**Step 4: Solve for 'a'**
Adding \(5\) to both sides:
\[ 7a = 30 + 5 = 35 \]
Dividing by \(7\):
\[ a = \frac{35}{7} = 5 \]

### Final Answers
(i) \(a = 12\)
(ii) \(a = 5\)