If `2x-3y=5`, what is the value of `4x^(2)-12xy+9y^(2)`?

Answer

To find the value of \( 4x^2 - 12xy + 9y^2 \) given that \( 2x - 3y = 5 \), we can follow these steps:

### Step 1: Recognize the Expression
The expression \( 4x^2 - 12xy + 9y^2 \) can be rewritten as \( (2x - 3y)^2 \) using the identity:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
Here, \( a = 2x \) and \( b = 3y \).

### Step 2: Apply the Identity
Using the identity, we can express:
\[ (2x - 3y)^2 = (2x)^2 - 2(2x)(3y) + (3y)^2 \]
This simplifies to:
\[ 4x^2 - 12xy + 9y^2 \]

### Step 3: Substitute the Given Value
Now, we know from the problem statement that \( 2x - 3y = 5 \). Therefore, we can substitute this into our expression:
\[ (2x - 3y)^2 = 5^2 \]

### Step 4: Calculate the Square
Calculating the square gives:
\[ (2x - 3y)^2 = 25 \]

### Conclusion
Thus, we find that:
\[ 4x^2 - 12xy + 9y^2 = 25 \]

### Final Answer
The value of \( 4x^2 - 12xy + 9y^2 \) is \( 25 \).

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