If x and y are real numbers, then the least possible value of `4(x-2)^(2)+(y-3)^(2)-2(x-3)^(2)` is

Options

A. 3
B. `-4`
C. 1
D. `-8`

Answer

To find the least possible value of the expression \(4(x-2)^{2} + (y-3)^{2} - 2(x-3)^{2}\), we will simplify and analyze the expression step by step.

### Step 1: Simplify the Expression
Start with the given expression:
\[ E = 4(x-2)^{2} + (y-3)^{2} - 2(x-3)^{2} \]

### Step 2: Expand the Squares
Expand the squares in the expression:
\[ E = 4((x-2)^{2}) + (y-3)^{2} - 2((x-3)^{2}) \]
Calculating each square:
\[ (x-2)^{2} = x^{2} - 4x + 4 \]
\[ (x-3)^{2} = x^{2} - 6x + 9 \]
Substituting these back into the expression:
\[ E = 4(x^{2} - 4x + 4) + (y-3)^{2} - 2(x^{2} - 6x + 9) \]
\[ E = 4x^{2} - 16x + 16 + (y-3)^{2} - 2x^{2} + 12x - 18 \]

### Step 3: Combine Like Terms
Now, combine the like terms:
\[ E = (4x^{2} - 2x^{2}) + (-16x + 12x) + (16 - 18) + (y-3)^{2} \]
\[ E = 2x^{2} - 4x - 2 + (y-3)^{2} \]

### Step 4: Minimize the Expression
To minimize \(E\), we can minimize each part separately. First, consider the term \( (y-3)^{2} \):
- The minimum value occurs when \(y = 3\), giving \( (y-3)^{2} = 0\).

Now, substitute \(y = 3\) into the expression:
\[ E = 2x^{2} - 4x - 2 \]

### Step 5: Complete the Square for \(x\)
Next, we need to minimize \(2x^{2} - 4x - 2\). We can complete the square:
\[ E = 2(x^{2} - 2x) - 2 \]
To complete the square:
\[ x^{2} - 2x = (x-1)^{2} - 1 \]
Thus,
\[ E = 2((x-1)^{2} - 1) - 2 \]
\[ E = 2(x-1)^{2} - 2 - 2 \]
\[ E = 2(x-1)^{2} - 4 \]

### Step 6: Find the Minimum Value
The term \(2(x-1)^{2}\) is minimized when \(x = 1\), giving:
\[ 2(0) - 4 = -4 \]

### Conclusion
The least possible value of the expression \(E\) is:
\[ \boxed{-4} \]