The term , that should be added to ` ( 4 x ^(2) + 8x)` so that resulting expression be a perfect square, is

Options

A. 2
B. 4
C. 2x
D. 1

Answer

To determine the term that should be added to the expression \( 4x^2 + 8x \) so that the resulting expression is a perfect square, we can follow these steps:

### Step 1: Identify the coefficients
The given expression is \( 4x^2 + 8x \). Here, we can identify:
- The coefficient of \( x^2 \) is \( 4 \).
- The coefficient of \( x \) is \( 8 \).

### Step 2: Rewrite the quadratic expression
We can factor out the coefficient of \( x^2 \) from the first two terms:
\[ 4(x^2 + 2x) \]

### Step 3: Complete the square
To complete the square for the expression \( x^2 + 2x \), we need to find the term that makes it a perfect square trinomial. The formula for completing the square is:
\[ x^2 + 2ax + a^2 \]
Here, \( a \) is half of the coefficient of \( x \).

In our case, the coefficient of \( x \) is \( 2 \), so:
\[ a = \frac{2}{2} = 1 \]
Now, we need to add \( a^2 = 1^2 = 1 \) to the expression \( x^2 + 2x \).

### Step 4: Add the term to the expression
Since we factored out \( 4 \) in step 2, we need to add \( 4 \times 1 = 4 \) to the original expression to maintain equality:
\[ 4(x^2 + 2x + 1) = 4(x + 1)^2 \]

### Conclusion
Thus, the term that should be added to \( 4x^2 + 8x \) to make it a perfect square is \( 4 \).

### Final Answer
The term that should be added is \( 4 \).

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