Solve `9x^(2)-9(a+b)x+5ab+2b^(2)=0`.

Answer

To solve the quadratic equation \(9x^2 - 9(a+b)x + (5ab + 2b^2) = 0\), we will use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 1: Identify coefficients
From the equation \(9x^2 - 9(a+b)x + (5ab + 2b^2) = 0\), we identify:
- \(a = 9\)
- \(b = -9(a+b)\)
- \(c = 5ab + 2b^2\)

### Step 2: Substitute coefficients into the formula
Now we can substitute these values into the quadratic formula:

\[ x = \frac{-(-9(a+b)) \pm \sqrt{(-9(a+b))^2 - 4 \cdot 9 \cdot (5ab + 2b^2)}}{2 \cdot 9} \]

### Step 3: Simplify the expression
This simplifies to:

\[ x = \frac{9(a+b) \pm \sqrt{(9(a+b))^2 - 36(5ab + 2b^2)}}{18} \]

### Step 4: Calculate \(b^2 - 4ac\)
Now we calculate \(b^2 - 4ac\):

\[ (9(a+b))^2 = 81(a+b)^2 = 81(a^2 + 2ab + b^2) \]

And:

\[ 36(5ab + 2b^2) = 180ab + 72b^2 \]

Now substituting these into the discriminant:

\[ b^2 - 4ac = 81(a^2 + 2ab + b^2) - (180ab + 72b^2) \]

### Step 5: Combine like terms
Combining the terms gives:

\[ 81a^2 + 162ab + 81b^2 - 180ab - 72b^2 = 81a^2 - 18ab + 9b^2 \]

### Step 6: Factor the discriminant
We can factor this expression:

\[ 81a^2 - 18ab + 9b^2 = (9a - 3b)^2 \]

### Step 7: Substitute back into the formula
Now substituting back into the quadratic formula gives:

\[ x = \frac{9(a+b) \pm (9a - 3b)}{18} \]

### Step 8: Split into two cases
This results in two cases:

1. \(x_1 = \frac{9(a+b) + (9a - 3b)}{18} = \frac{18a + 6b}{18} = a + \frac{b}{3}\)
2. \(x_2 = \frac{9(a+b) - (9a - 3b)}{18} = \frac{6b}{18} = \frac{b}{3}\)

### Final Answer:
Thus, the solutions are:

\[ x_1 = a + \frac{b}{3}, \quad x_2 = \frac{b}{3} \]