Question

What number must be added to the expression 16a²-12a to make it a perfect square?

• A. \(\frac{9}{4}\)
• B. \(\frac{11}{2}\)
• C. \(\frac{13}{2}\)
• D. 16

Answer 1

The Correct Option is A

To make the expression \(16a^2 - 12a\) a perfect square, we need to rewrite it in the form \((ma + n)^2 = m^2a^2 + 2mna + n^2\), where \(m\) and \(n\) are numbers.
Step 1: Start by factoring the coefficient of \(a^2\), giving us \(16a^2\). The square root of \(16a^2\) is \(4a\). Therefore, \(m = 4\).
Step 2: Rewriting the expression, \(16a^2 - 12a = (4a)^2 - 12a\).
Step 3: Compare it with \((ma + n)^2 = m^2a^2 + 2mna + n^2\), which gives \(2mna = -12a\). Substitute \(m = 4\): \(2(4)(n)a = -12a\), resulting in \(8na = -12a\). Thus, \(8n = -12\).
Step 4: Solve the equation: \(n = -\frac{12}{8} = -\frac{3}{2}\).
Step 5: Calculate the perfect square addition: Since \((4a - \frac{3}{2})^2 = (4a)^2 - 2\cdot 4\cdot \frac{3}{2} \cdot a + (\frac{3}{2})^2\), we find \((\frac{3}{2})^2 = \frac{9}{4}\).
Step 6: The number to be added is \(\frac{9}{4}\) to make the expression a perfect square. Therefore, the correct option is \(\frac{9}{4}\).