Solve ` 9x^(2)-12x +25=0`

Answer

To solve the quadratic equation \( 9x^2 - 12x + 25 = 0 \), we will use the quadratic formula:

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

### Step 1: Identify coefficients
From the equation \( 9x^2 - 12x + 25 = 0 \), we can identify the coefficients:
- \( a = 9 \)
- \( b = -12 \)
- \( c = 25 \)

### Step 2: Calculate the discriminant
Next, we calculate the discriminant \( D = b^2 - 4ac \):

\[ D = (-12)^2 - 4 \cdot 9 \cdot 25 \]
\[ D = 144 - 900 \]
\[ D = -756 \]

### Step 3: Substitute into the quadratic formula
Now, we substitute \( a \), \( b \), and \( D \) into the quadratic formula:

\[ x = \frac{-(-12) \pm \sqrt{-756}}{2 \cdot 9} \]
\[ x = \frac{12 \pm \sqrt{-756}}{18} \]

### Step 4: Simplify the square root of the negative number
Since \( \sqrt{-756} \) involves a negative number, we can express it using \( i \) (where \( i = \sqrt{-1} \)):

\[ \sqrt{-756} = \sqrt{756} \cdot i \]

Next, we simplify \( \sqrt{756} \). We can factor \( 756 \):

\[ 756 = 36 \cdot 21 \]
\[ \sqrt{756} = \sqrt{36 \cdot 21} = \sqrt{36} \cdot \sqrt{21} = 6\sqrt{21} \]

### Step 5: Substitute back into the equation
Now we substitute \( \sqrt{-756} \) back into our equation:

\[ x = \frac{12 \pm 6\sqrt{21}i}{18} \]

### Step 6: Simplify the fraction
We can simplify the fraction:

\[ x = \frac{12}{18} \pm \frac{6\sqrt{21}i}{18} \]
\[ x = \frac{2}{3} \pm \frac{\sqrt{21}}{3}i \]

### Final Answer
Thus, the solutions to the equation \( 9x^2 - 12x + 25 = 0 \) are:

\[ x = \frac{2}{3} + \frac{\sqrt{21}}{3}i \quad \text{and} \quad x = \frac{2}{3} - \frac{\sqrt{21}}{3}i \]

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