Question

Find the square root of $4a^2 + 9b^2 + c^2 + 12ab - 6bc - 4ac$.

• A. \(2a + b - 3c \)
• B. \(2a - 3b - c \)
• C. \(2a - 3b + c \)
• D. \(2a + 3b - c \)
• E. \(2a + b - 2c \)

Hint:

To identify perfect squares: - Compare with $(x+y+z)^2$ - Check signs carefully for cross terms

Answer 1

The Correct Option is D

Concept: We use the identity: \[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx \] Match the given expression with a perfect square.
Step 1: Compare with $(2a + 3b - c)^2$.
\[ (2a + 3b - c)^2 = (2a)^2 + (3b)^2 + (-c)^2 + 2(2a)(3b) + 2(3b)(-c) + 2(2a)(-c) \] \[ = 4a^2 + 9b^2 + c^2 + 12ab - 6bc - 4ac \]

Step 2: Match with given expression.
The expression exactly matches: \[ 4a^2 + 9b^2 + c^2 + 12ab - 6bc - 4ac \]

Step 3: Final answer.
\[ \sqrt{4a^2 + 9b^2 + c^2 + 12ab - 6bc - 4ac} = 2a + 3b - c \]