The factors of `9a ^(2) - 6 sqrt5 a + 5` are

Options

A. `(3a + sqrt5) ( 3a - sqrt5)`
B. `( 3a - 5) (3a - 5)`
C. `( 3a - sqrt5) ( 3a - sqrt5)`
D. `(3a + sqrt5) ( 3 a - 5)`

Answer

To factor the expression \(9a^2 - 6\sqrt{5}a + 5\), we can follow these steps:

### Step 1: Identify the coefficients
The given expression is a quadratic in the form \(ax^2 + bx + c\), where:
- \(a = 9\)
- \(b = -6\sqrt{5}\)
- \(c = 5\)

### Step 2: Calculate the product and sum
We need to find two numbers that multiply to \(ac\) (which is \(9 \times 5 = 45\)) and add up to \(b\) (which is \(-6\sqrt{5}\)).

### Step 3: Determine the two numbers
We can express \(45\) as \(3\sqrt{5} \times 3\sqrt{5}\) because:
- \(3 \times 3 = 9\)
- \(\sqrt{5} \times \sqrt{5} = 5\)
Thus, \(3\sqrt{5} + 3\sqrt{5} = 6\sqrt{5}\). Since we need a negative sum, we can use \(-3\sqrt{5}\) and \(-3\sqrt{5}\).

### Step 4: Rewrite the middle term
We can rewrite the expression as:
\[ 9a^2 - 3\sqrt{5}a - 3\sqrt{5}a + 5 \]

### Step 5: Group the terms
Now, we will group the terms:
\[ (9a^2 - 3\sqrt{5}a) + (-3\sqrt{5}a + 5) \]

### Step 6: Factor by grouping
Now, we factor out the common factors from each group:
1. From the first group \(9a^2 - 3\sqrt{5}a\), we can factor out \(3a\):
\[ 3a(3a - \sqrt{5}) \]
2. From the second group \(-3\sqrt{5}a + 5\), we can factor out \(-\sqrt{5}\):
\[ -\sqrt{5}(3a - \sqrt{5}) \]

So, we have:
\[ 3a(3a - \sqrt{5}) - \sqrt{5}(3a - \sqrt{5}) \]

### Step 7: Combine the factors
Now we can combine the two groups:
\[ (3a - \sqrt{5})(3a - \sqrt{5}) \]

### Step 8: Write the final factorized form
Thus, the factorized form of the expression is:
\[ (3a - \sqrt{5})^2 \]

### Conclusion
The factors of \(9a^2 - 6\sqrt{5}a + 5\) are:
\[ 3a - \sqrt{5} \quad \text{and} \quad 3a - \sqrt{5} \]