What number must be added to the expression `16a^(2)-12a` to make it a perfect square ?

Options

A. `(9)/(4)`
B. `(11)/(2)`
C. `(13)/(2)`
D. `16`

Answer

To determine what number must be added to the expression \(16a^2 - 12a\) to make it a perfect square, we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ 16a^2 - 12a \]

### Step 2: Rewrite the expression in a suitable form
We can factor out the coefficient of \(a^2\) from the first two terms:
\[ 16a^2 - 12a = 4a(4a - 3) \]
However, for our purpose, we will focus on completing the square for the quadratic part.

### Step 3: Completing the square
To complete the square for the expression \(16a^2 - 12a\), we will use the formula for a perfect square trinomial, which is:
\[ (A - B)^2 = A^2 - 2AB + B^2 \]
Here, we identify \(A\) and \(B\) such that:
- \(A^2 = 16a^2\) implies \(A = 4a\)
- We need to find \(B\) such that \(-2AB = -12a\)

### Step 4: Find \(B\)
From \(-2AB = -12a\):
\[ -2(4a)B = -12a \]
Dividing both sides by \(-2a\):
\[ 4B = 12 \implies B = 3 \]

### Step 5: Write the perfect square
Now we can write the expression as:
\[ (4a - 3)^2 \]
This expands to:
\[ (4a)^2 - 2(4a)(3) + 3^2 = 16a^2 - 24a + 9 \]

### Step 6: Determine the number to add
We need to compare this with our original expression \(16a^2 - 12a\). The term we have in the perfect square expansion is \(-24a\), but we only have \(-12a\) in our original expression.

To make \(16a^2 - 12a\) a perfect square, we need to add the difference between the two:
\[ (16a^2 - 24a + 9) - (16a^2 - 12a) = -24a + 12a + 9 = -12a + 9 \]
Thus, we need to add \(9\) to the expression.

### Step 7: Final answer
The number that must be added to the expression \(16a^2 - 12a\) to make it a perfect square is:
\[ \frac{9}{4} \]

### Summary
The answer is \(\frac{9}{4}\).