Solve the following quations by using qardratic formula:
`36x^(2)-12 ax+(a^(2)-b^(2))=0`

Answer

To solve the quadratic equation \( 36x^{2} - 12ax + (a^{2} - b^{2}) = 0 \) using the quadratic formula, we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \( ax^{2} + bx + c = 0 \). From the given equation, we can identify:
- \( a = 36 \)
- \( b = -12a \)
- \( c = a^{2} - b^{2} \)

### Step 2: Write the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]

### Step 3: Substitute the values of \( a \), \( b \), and \( c \) into the formula
Substituting the values we found:
\[ x = \frac{-(-12a) \pm \sqrt{(-12a)^{2} - 4 \cdot 36 \cdot (a^{2} - b^{2})}}{2 \cdot 36} \]

### Step 4: Simplify the equation
Calculating \( -(-12a) \):
\[ x = \frac{12a \pm \sqrt{144a^{2} - 4 \cdot 36 \cdot (a^{2} - b^{2})}}{72} \]

Calculating \( 4 \cdot 36 \):
\[ x = \frac{12a \pm \sqrt{144a^{2} - 144(a^{2} - b^{2})}}{72} \]

Distributing \( -144 \):
\[ x = \frac{12a \pm \sqrt{144a^{2} - 144a^{2} + 144b^{2}}}{72} \]

This simplifies to:
\[ x = \frac{12a \pm \sqrt{144b^{2}}}{72} \]

### Step 5: Further simplify the square root
Since \( \sqrt{144b^{2}} = 12b \):
\[ x = \frac{12a \pm 12b}{72} \]

### Step 6: Factor out the common terms
Factoring out \( 12 \):
\[ x = \frac{12(a \pm b)}{72} \]

### Step 7: Simplify the fraction
This simplifies to:
\[ x = \frac{a \pm b}{6} \]

### Final Solutions
Thus, the solutions for the quadratic equation are:
\[ x = \frac{a + b}{6} \quad \text{and} \quad x = \frac{a - b}{6} \]