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

To make the expression \(16a^2 – 12a\) a perfect square, we can complete the square by adding a term to it.

First, let’s factor out the common factor of 4:

\[16a^2 – 12a = 4(4a^2 – 3a)\]

Now, we want to complete the square for the expression inside the parentheses. The general form for a perfect square is \((x – y)^2 = x^2 – 2xy + y^2\). In our case, we have \(4a^2 – 3a\), so we can compare it to \(x^2 – 2xy\) to find the missing \(y^2\) term:

Comparing \(4a^2 – 3a\) to \(x^2 – 2xy\), we get:

– \(x = 2a\) (since \(x^2 = 4a^2\))
– \(-2xy = -3a\), so \(y = \frac{3}{4}\) (since \(x = 2a\))

Therefore, the missing \(y^2\) term is \(\left(\frac{3}{4}\right)^2 = \frac{9}{16}\).

However, remember that we factored out a 4 earlier, so we need to multiply this term by 4 to add it to the original expression:

\[4 \times \frac{9}{16} = \frac{9}{4}\]

So, the number that must be added to the expression \(16a^2 – 12a\) to make it a perfect square is \(\frac{9}{4}\).