`9 x^(2) + 25 - 30x` can be expressed as the square of

Options

A. `-3x - 5 `
B. `3 x + 5`
C. `3 x - 5`
D. `3 x^(2) - 25`

Answer

To express the expression \( 9x^2 + 25 - 30x \) as a square, we can follow these steps:

### Step 1: Rearrange the expression
Start by rearranging the expression to group the terms appropriately:
\[ 9x^2 - 30x + 25 \]

### Step 2: Identify the form of a perfect square
Recall the formula for a perfect square:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
We need to identify \( a \) and \( b \) such that:
- \( a^2 = 9x^2 \)
- \( b^2 = 25 \)
- \( -2ab = -30x \)

### Step 3: Solve for \( a \) and \( b \)
From \( a^2 = 9x^2 \), we find:
\[ a = 3x \]
From \( b^2 = 25 \), we find:
\[ b = 5 \]

### Step 4: Verify the middle term
Now, we need to check if \( -2ab = -30x \):
\[ -2(3x)(5) = -30x \]
This is correct, so we can express the original expression as:
\[ (3x - 5)^2 \]

### Conclusion
Thus, the expression \( 9x^2 + 25 - 30x \) can be expressed as the square of:
\[ (3x - 5) \]

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