Simplify : `(3y)^(2) +x^(2)- (2y)^(2)`

Options

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

Answer

To simplify the expression \( (3y)^{2} + x^{2} - (2y)^{2} \), we will follow these steps:

### Step 1: Expand the squares
We start by expanding each squared term in the expression.

\[ (3y)^{2} = 3^{2} \cdot y^{2} = 9y^{2} \]
\[ (2y)^{2} = 2^{2} \cdot y^{2} = 4y^{2} \]

Now, substituting these back into the original expression gives us:

\[ 9y^{2} + x^{2} - 4y^{2} \]

### Step 2: Combine like terms
Next, we combine the like terms, which are the terms involving \( y^{2} \).

\[ 9y^{2} - 4y^{2} = (9 - 4)y^{2} = 5y^{2} \]

So, the expression now simplifies to:

\[ 5y^{2} + x^{2} \]

### Step 3: Write the final simplified expression
The final simplified expression is:

\[ x^{2} + 5y^{2} \]

### Final Answer
Thus, the simplified form of the expression \( (3y)^{2} + x^{2} - (2y)^{2} \) is:

\[ x^{2} + 5y^{2} \]

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