If a = 2, b = 3 and c = -2, find the value of `a^(2)+b^(2)+c^(2)-2b-2bc-2ca+3abc`

Answer

To solve the expression \( a^{2} + b^{2} + c^{2} - 2b - 2bc - 2ca + 3abc \) given that \( a = 2 \), \( b = 3 \), and \( c = -2 \), we will substitute the values step by step.

### Step 1: Substitute the values of a, b, and c into the expression.

Given:
- \( a = 2 \)
- \( b = 3 \)
- \( c = -2 \)

The expression becomes:
\[ (2)^{2} + (3)^{2} + (-2)^{2} - 2(3) - 2(3)(-2) - 2(2)(-2) + 3(2)(3)(-2) \]

### Step 2: Calculate each term.

1. Calculate \( a^{2} \):
\[ (2)^{2} = 4 \]

2. Calculate \( b^{2} \):
\[ (3)^{2} = 9 \]

3. Calculate \( c^{2} \):
\[ (-2)^{2} = 4 \]

4. Calculate \( -2b \):
\[ -2(3) = -6 \]

5. Calculate \( -2bc \):
\[ -2(3)(-2) = 12 \]

6. Calculate \( -2ca \):
\[ -2(2)(-2) = 8 \]

7. Calculate \( 3abc \):
\[ 3(2)(3)(-2) = -36 \]

### Step 3: Combine all the calculated values.

Now, substitute all the calculated values back into the expression:
\[ 4 + 9 + 4 - 6 + 12 + 8 - 36 \]

### Step 4: Perform the addition and subtraction.

1. Start with \( 4 + 9 = 13 \)
2. Then \( 13 + 4 = 17 \)
3. Next, \( 17 - 6 = 11 \)
4. Then \( 11 + 12 = 23 \)
5. Next, \( 23 + 8 = 31 \)
6. Finally, \( 31 - 36 = -5 \)

### Final Answer:
The value of the expression is \( -5 \).

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