Question

For what values of \(k\) are the roots of the quadratic equation \(9x^2 + 3kx + 4 = 0\) real and equal?

• A. \(\pm 4\)
• B. \(\pm 7\)
• C. \(\pm 9\)
• D. \(\pm 6\)

Hint:

“Real and equal roots” \(\Rightarrow\) discriminant \(D=0\); “real and distinct” \(\Rightarrow D>0\); “no real roots” \(\Rightarrow D<0\).

Answer 1

The Correct Option is A

Step 1: Use the discriminant condition for equal real roots.
For \(ax^2+bx+c=0\), roots are real and equal when \(D=b^2-4ac=0\).
Here \(a=9\), \(b=3k\), \(c=4\).
Step 2: Set the discriminant to zero and solve for \(k\).
\[ D=(3k)^2-4\cdot 9\cdot 4 = 9k^2-144=0 \;\Rightarrow\; 9k^2=144 \;\Rightarrow\; k^2=16 \;\Rightarrow\; k=\pm 4. \]
Step 3: Conclude.
Hence, the required values of \(k\) are \(\boxed{\pm 4}\).