If `A = 9t^(2) + 14xt - 3x^(2), B = 16xt + 4x^(2) - 5t^(2) " and " C = -7x^(2) - 19xt - 8t^(2)`, Then find the value of `B - A - C`.

Options

A. `-6t^(2) + 21xt + 14x^(2)`
B. `6t^(2) - 21xt - 14x^(2)`
C. `14x^(2) +15xt - 3t^(2)`
D. `17x^(2) + 21xt + 19t^(2)`

Answer

To find the value of \( B - A - C \), we will first write down the expressions for \( A \), \( B \), and \( C \):

1. **Given Expressions**:
- \( A = 9t^2 + 14xt - 3x^2 \)
- \( B = 16xt + 4x^2 - 5t^2 \)
- \( C = -7x^2 - 19xt - 8t^2 \)

2. **Substituting the Values**:
We need to calculate \( B - A - C \):
\[ B - A - C = (16xt + 4x^2 - 5t^2) - (9t^2 + 14xt - 3x^2) - (-7x^2 - 19xt - 8t^2) \]

3. **Distributing the Negative Signs**:
When we subtract \( A \) and \( C \), we need to distribute the negative signs:
\[ B - A - C = 16xt + 4x^2 - 5t^2 - 9t^2 - 14xt + 3x^2 + 7x^2 + 19xt + 8t^2 \]

4. **Combining Like Terms**:
Now, we will combine the like terms:
- For \( xt \): \( 16xt - 14xt + 19xt = (16 - 14 + 19)xt = 21xt \)
- For \( x^2 \): \( 4x^2 + 3x^2 + 7x^2 = (4 + 3 + 7)x^2 = 14x^2 \)
- For \( t^2 \): \( -5t^2 - 9t^2 + 8t^2 = (-5 - 9 + 8)t^2 = -6t^2 \)

5. **Final Expression**:
Putting it all together, we have:
\[ B - A - C = 21xt + 14x^2 - 6t^2 \]

Thus, the final answer is:
\[ B - A - C = 21xt + 14x^2 - 6t^2 \]