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
In the given quadratic equation, we can identify the coefficients as follows:
- \( a = 9 \)
- \( b = -9(a+b) \)
- \( c = 5ab + 2b^2 \)

### Step 2: Substitute coefficients into the quadratic formula
Now, we substitute these coefficients 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
First, simplify \( -(-9(a+b)) \):

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

Next, calculate \( (9(a+b))^2 \):

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

Now, simplify \( 36(5ab + 2b^2) \):

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

### Step 4: Combine under the square root
Now we can combine these results:

\[ x = \frac{9(a+b) \pm \sqrt{81(a+b)^2 - (180ab + 72b^2)}}{18} \]

### Step 5: Further simplify the expression under the square root
Now, we need to simplify \( 81(a+b)^2 - (180ab + 72b^2) \):

Expanding \( 81(a+b)^2 \):

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

So we have:

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

### Step 6: Substitute back into the equation
Now, substituting back, we have:

\[ x = \frac{9(a+b) \pm \sqrt{81a^2 - 18ab + 9b^2}}{18} \]

### Step 7: Factor out common terms
Notice that \( \sqrt{81a^2 - 18ab + 9b^2} = \sqrt{(9a - 3b)^2} = 9a - 3b \):

Thus, we can write:

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

### Step 8: Solve for the two possible values of \( x \)
Now we can solve for the two cases:

1. **Case 1**: \( x = \frac{9(a+b) + (9a - 3b)}{18} \)
\[ = \frac{18a + 6b}{18} = a + \frac{b}{3} \]

2. **Case 2**: \( x = \frac{9(a+b) - (9a - 3b)}{18} \)
\[ = \frac{12b}{18} = \frac{2b}{3} \]

### Final Solution
Thus, the solutions for the quadratic equation are:

\[ x = a + \frac{b}{3} \quad \text{and} \quad x = \frac{2b}{3} \]