Question

If the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are equal, then the value of \( c \) is:

• A. \( \frac{a^2}{4b} \)
• B. \( \frac{a^2}{b} \)
• C. \( \frac{b^2}{a} \)
• D. \( \frac{b^2}{4a} \)

Hint:

For a quadratic equation to have equal roots, its discriminant must be zero. The discriminant is \( \Delta = b^2 - 4ac \).

Answer 1

The Correct Option is D

For the quadratic equation \( ax^2 + bx + c = 0 \), the condition for equal roots is that the discriminant \( \Delta \) must be zero. The discriminant for the quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ \Delta = b^2 - 4ac \]
Step 1: Set the discriminant equal to zero for equal roots.
For equal roots, the discriminant must be zero, so: \[ b^2 - 4ac = 0 \]
Step 2: Solve for \( c \).
Rearranging the equation: \[ b^2 = 4ac \] Now, solving for \( c \): \[ c = \frac{b^2}{4a} \]
Step 3: Conclusion.
Therefore, the value of \( c \) is \( \frac{b^2}{4a} \).