Question

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

• A. $9/4$
• B. $13/2$
• C. $11/2$
• D. $16/9$
• E. $13/4$

Hint:

Completing the square is an essential algebraic manipulation tool. In fields like AI, quadratic optimization and cost function minimization frequently rely on rearranging polynomial equations into perfect squares.

Answer 1

The Correct Option is A

Step 1: Match the expression to the standard binomial square formula.
The formula is $(x - y)^{2} = x^{2} - 2xy + y^{2}$.
Given terms: $16a^{2} - 12a$.
Let $x^{2} = 16a^{2}$, which means $x = 4a$.

Step 2: Solve for $y$.
The middle term is $-2xy = -12a$.
Substitute $x = 4a$: $-2(4a)y = -12a \Rightarrow -8ay = -12a \Rightarrow y = \frac{12}{8} = \frac{3}{2}$.

Step 3: Find the missing term $y^{2}$.
To complete the square, we must add $y^{2}$.
$y^{2} = (\frac{3}{2})^{2} = \frac{9}{4}$.