Question

What values of k will make 9x²+3kx+4 a perfect square?

• A. ± 1
• B. ±2
• C. ± 3
• D. ± 4
• E. ± 5

Answer 1

The Correct Option is D

Step 1: Understand the given quadratic expression.
The quadratic expression is: \[ 9x^2 + 3kx + 4 \] We need to find the values of \( k \) that will make this expression a perfect square.

Step 2: General form of a perfect square.
For a quadratic expression to be a perfect square, it must be expressible as the square of a binomial, i.e., \[ (ax + b)^2 = a^2x^2 + 2abx + b^2 \] Comparing this with the given quadratic expression, we can see that: \[ a^2 = 9 \quad \text{and} \quad b^2 = 4 \] Thus, \( a = 3 \) (since \( a^2 = 9 \), so \( a = 3 \)) and \( b = 2 \) or \( b = -2 \) (since \( b^2 = 4 \), so \( b = 2 \) or \( b = -2 \)).

Step 3: Compare the linear term.
The linear term of the perfect square expansion is: \[ 2abx \] In our case, \( a = 3 \), so: \[ 2ab = 2 \times 3 \times b = 6b \] Now compare this with the linear term of the given expression, which is \( 3kx \). For the expression to be a perfect square, we must have: \[ 6b = 3k \] Therefore, \( k = 2b \).

Step 4: Solve for \( k \).
Since \( b = 2 \) or \( b = -2 \), we have two possible values for \( k \): - If \( b = 2 \), then \( k = 2 \times 2 = 4 \). - If \( b = -2 \), then \( k = 2 \times (-2) = -4 \).

Step 5: Conclusion.
The values of \( k \) that make the expression a perfect square are \( k = 4 \) and \( k = -4 \).

Final Answer:
The correct option is (D): ± 4.