The equation `x^2-3(x-3)^2-2=0` has

Options

A. one minimum point
B. one maximum point
C. Both maximum and minimum points
D. None of the above

Answer

To solve the equation \( x^2 - 3(x - 3)^2 - 2 = 0 \), we will follow these steps:

### Step 1: Expand the equation
Start by expanding the term \( (x - 3)^2 \):
\[ (x - 3)^2 = x^2 - 6x + 9 \]
Now substitute this back into the equation:
\[ x^2 - 3(x^2 - 6x + 9) - 2 = 0 \]

### Step 2: Distribute the -3
Distributing \(-3\) gives:
\[ x^2 - 3x^2 + 18x - 27 - 2 = 0 \]

### Step 3: Combine like terms
Combine the \(x^2\) terms and the constant terms:
\[ -2x^2 + 18x - 29 = 0 \]

### Step 4: Rearranging the equation
To make it easier to work with, we can multiply through by -1:
\[ 2x^2 - 18x + 29 = 0 \]

### Step 5: Use the quadratic formula
Now we will use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -18 \), and \( c = 29 \):
1. Calculate the discriminant:
\[ b^2 - 4ac = (-18)^2 - 4 \cdot 2 \cdot 29 = 324 - 232 = 92 \]
2. Now substitute into the quadratic formula:
\[ x = \frac{18 \pm \sqrt{92}}{4} \]

### Step 6: Simplify the square root
The square root of 92 can be simplified:
\[ \sqrt{92} = \sqrt{4 \cdot 23} = 2\sqrt{23} \]
So the equation becomes:
\[ x = \frac{18 \pm 2\sqrt{23}}{4} = \frac{9 \pm \sqrt{23}}{2} \]

### Step 7: Final solutions
Thus, the solutions to the equation are:
\[ x = \frac{9 + \sqrt{23}}{2} \quad \text{and} \quad x = \frac{9 - \sqrt{23}}{2} \]

### Conclusion
The equation \( x^2 - 3(x - 3)^2 - 2 = 0 \) has two solutions:
\[ x = \frac{9 + \sqrt{23}}{2} \quad \text{and} \quad x = \frac{9 - \sqrt{23}}{2} \]

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