Number of real solution(s) of the equation `|x-3|^(3x^2-10x+3)=1` is :

Options

A. exactly four
B. exactly three
C. exactly two
D. exactly one

Answer

To find the number of real solutions for the equation \( |x-3|^{(3x^2 - 10x + 3)} = 1 \), we can analyze the equation step by step.

### Step 1: Understand the conditions for the equation to hold true.
The equation \( a^b = 1 \) can be true under the following conditions:
1. \( a = 1 \) (where \( a \) is not equal to 0)
2. \( a = -1 \) and \( b \) is an even integer (where \( a \) is not equal to 0)
3. \( b = 0 \) (where \( a \) is not equal to 0)

### Step 2: Analyze the first condition \( |x-3| = 1 \).
Setting \( |x-3| = 1 \) gives us two cases:
1. \( x - 3 = 1 \) → \( x = 4 \)
2. \( x - 3 = -1 \) → \( x = 2 \)

### Step 3: Check the exponent \( 3x^2 - 10x + 3 \) for these values.
Now we need to evaluate the exponent \( 3x^2 - 10x + 3 \) for \( x = 2 \) and \( x = 4 \).

- For \( x = 2 \):
\[ 3(2^2) - 10(2) + 3 = 3(4) - 20 + 3 = 12 - 20 + 3 = -5 \quad (\text{not } 0) \]

- For \( x = 4 \):
\[ 3(4^2) - 10(4) + 3 = 3(16) - 40 + 3 = 48 - 40 + 3 = 11 \quad (\text{not } 0) \]

### Step 4: Analyze the second condition \( |x-3| = -1 \).
Since the absolute value cannot be negative, this condition does not yield any valid solutions.

### Step 5: Analyze the third condition \( 3x^2 - 10x + 3 = 0 \).
Now we set the exponent to zero:
\[ 3x^2 - 10x + 3 = 0 \]
We can factor this quadratic equation:
\[ 3x^2 - 9x - x + 3 = 0 \\ 3x(x - 3) - 1(x - 3) = 0 \\ (3x - 1)(x - 3) = 0 \]
This gives us:
1. \( x = 3 \) (not valid since it makes \( |x-3| = 0 \))
2. \( x = \frac{1}{3} \) (valid)

### Step 6: Summarize the valid solutions.
The valid solutions we found are:
1. \( x = 2 \)
2. \( x = 4 \)
3. \( x = \frac{1}{3} \)

### Conclusion:
Thus, the total number of real solutions for the equation \( |x-3|^{(3x^2 - 10x + 3)} = 1 \) is **3**.

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