Question

If the quadratic equation \(9x^2 + 8kx + 16 = 0\) has real and equal roots, then the value of k is

• A. 3
• B. –3
• C. –4
• D. \(\pm 3\)

Hint:

When taking the square root on both sides of an algebraic equation like \(k^2 = 9\), always remember to consider both the positive and negative roots.
A common error is to only write the positive value \(k = 3\), which would lead to selecting the incorrect option (A).
Always double-check all options before finalizing your answer!

Answer 1

The Correct Option is D

Step 1: Set up the equal roots.
For \(9x^2 + 8kx + 16 = 0\) to have real and equal roots, both roots must be the same number, say \(x = r\).
Step 2: Use the sum and product of roots.
Sum of roots \(= r + r = 2r = -\frac{8k}{9}\), and product of roots \(= r \times r = r^2 = \frac{16}{9}\).
Step 3: Solve for r, then for k.
From the product, \(r^2 = \frac{16}{9}\), so \(r = \pm\frac{4}{3}\). Using the sum relation \(2r = -\frac{8k}{9}\), we get \(k = -\frac{9r}{4}\). Putting \(r = \frac{4}{3}\) gives \(k = -3\), and putting \(r = -\frac{4}{3}\) gives \(k = 3\).
Step 4: Write the final answer.
So \(k = \pm 3\), which matches option (D).
\[ \boxed{k = \pm 3} \]