Question

The value of k for which the equation $kx^2 - 6x - 4 = 0$ has real and equal roots, is

• A. $\frac{9}{4}$
• B. $-4$
• C. $-\frac{9}{4}$
• D. $-2$

Hint:

For any quadratic equation $ax^2 + bx + c = 0$ to have real and equal roots, remember the direct condition $b^2 = 4ac$.
This allows you to quickly set up the equation and solve for the unknown parameter without extra intermediate steps.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a quadratic equation $kx^2 - 6x - 4 = 0$.
We need to find the value of the constant $k$ such that the quadratic equation has real and equal roots.
The nature of the roots of any quadratic equation is determined by its discriminant.

Step 2: Key Formula or Approach:
For a standard quadratic equation $ax^2 + bx + c = 0$, the roots are real and equal if and only if the discriminant $D$ is equal to zero:
\[ D = b^2 - 4ac = 0 \]
We identify the coefficients $a$, $b$, and $c$ from the given equation and solve for $k$.

Step 3: Detailed Explanation:

• Identify the coefficients of the given quadratic equation $kx^2 - 6x - 4 = 0$:
\[ a = k, \quad b = -6, \quad c = -4 \]

• Set the discriminant $D$ to zero for real and equal roots:
\[ b^2 - 4ac = 0 \]

• Substitute the values of $a$, $b$, and $c$:
\[ (-6)^2 - 4(k)(-4) = 0 \]

• Simplify the equation:
\[ 36 + 16k = 0 \]

• Isolate the variable $k$:
\[ 16k = -36 \]
\[ k = -\frac{36}{16} \]

• Reduce the fraction to its simplest form by dividing both the numerator and denominator by 4:
\[ k = -\frac{9}{4} \]


Step 4: Final Answer:
The value of $k$ for which the equation has real and equal roots is $-\frac{9}{4}$.