`((9)/(4)x^(2)-1)-((3)/(2)x-1)^(2)`
The expression above is equivalent to

Options

A. `3x-2`
B. `-3x`
C. `(3)/(4)x-2`
D. `0`

Answer

To simplify the expression \(\frac{9}{4}x^2 - 1 - \left(\frac{3}{2}x - 1\right)^2\), we will follow these steps:

### Step 1: Expand the square term
We start with the expression:
\[ \frac{9}{4}x^2 - 1 - \left(\frac{3}{2}x - 1\right)^2 \]
Using the formula for expanding a square, \((a - b)^2 = a^2 - 2ab + b^2\), we identify \(a = \frac{3}{2}x\) and \(b = 1\). Thus, we have:
\[ \left(\frac{3}{2}x - 1\right)^2 = \left(\frac{3}{2}x\right)^2 - 2 \cdot \frac{3}{2}x \cdot 1 + 1^2 \]
Calculating each term:
\[ \left(\frac{3}{2}x\right)^2 = \frac{9}{4}x^2, \quad -2 \cdot \frac{3}{2}x \cdot 1 = -3x, \quad 1^2 = 1 \]
So, we can rewrite the square:
\[ \left(\frac{3}{2}x - 1\right)^2 = \frac{9}{4}x^2 - 3x + 1 \]

### Step 2: Substitute back into the expression
Now, we substitute this expansion back into the original expression:
\[ \frac{9}{4}x^2 - 1 - \left(\frac{9}{4}x^2 - 3x + 1\right) \]
This simplifies to:
\[ \frac{9}{4}x^2 - 1 - \frac{9}{4}x^2 + 3x - 1 \]

### Step 3: Combine like terms
Now, we combine the like terms:
- The \(\frac{9}{4}x^2\) terms cancel out:
\[ \frac{9}{4}x^2 - \frac{9}{4}x^2 = 0 \]
- The constants and \(x\) terms combine:
\[ -1 - 1 + 3x = 3x - 2 \]

### Final Result
Thus, the expression simplifies to:
\[ 3x - 2 \]

### Conclusion
The expression \(\frac{9}{4}x^2 - 1 - \left(\frac{3}{2}x - 1\right)^2\) is equivalent to \(3x - 2\).

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