Question

For what values of $k$ does the quadratic equation $9x^2 + kx + 1 = 0$ have equal roots?

• A. $6,-6$
• B. $9,-9$
• C. $2,3$
• D. $-2,3$

Hint:

Whenever a quadratic equation asks for:
• Equal roots $\Rightarrow D=0$
• Distinct roots $\Rightarrow D>0$
• No real roots $\Rightarrow D<0$ Always start with the discriminant formula: \[ D=b^2-4ac \]

Answer 1

The Correct Option is A

Concept: For a quadratic equation \[ ax^2 + bx + c = 0 \] the discriminant is \[ D = b^2 - 4ac \] The nature of roots depends on the discriminant:
• If $D > 0$, roots are real and distinct.
• If $D = 0$, roots are real and equal.
• If $D < 0$, roots are imaginary. Since the question asks for equal roots, we use: \[ D = 0 \]

Step 1: Identify the coefficients. Given equation: \[ 9x^2 + kx + 1 = 0 \] Comparing with \[ ax^2 + bx + c = 0 \] we get: \[ a = 9 \] \[ b = k \] \[ c = 1 \]

Step 2: Apply the equal roots condition. For equal roots: \[ b^2 - 4ac = 0 \] Substitute the values: \[ k^2 - 4(9)(1) = 0 \] \[ k^2 - 36 = 0 \]

Step 3: Solve the equation. Move 36 to the other side: \[ k^2 = 36 \] Take square root on both sides: \[ k = \pm \sqrt{36} \] \[ k = \pm 6 \] Thus, \[ k = 6 \quad \text{or} \quad k = -6 \] Hence, the correct option is \[ \boxed{(1)\ 6,-6} \]