Question

If the roots of the quadratic equation $ x^2 + 4x + k = 0 $ are real and equal, then the value of $ k $ is:

• A. 2
• B. 4
• C. 8
• D. 16

Hint:

Key Fact: A quadratic equation has equal roots if \( D = 0 \), i.e., \( b^2 = 4ac \)

Answer 1

The Correct Option is B

A quadratic equation \( ax^2 + bx + c = 0 \) has real and equal roots when the discriminant is zero. 

The discriminant formula is: \[ D = b^2 - 4ac \]

Given equation: \( x^2 + 4x + k = 0 \Rightarrow a = 1,\ b = 4,\ c = k \)

Apply the condition for equal roots: \[ D = 4^2 - 4(1)(k) = 16 - 4k = 0 \Rightarrow 4k = 16 \Rightarrow k = 4 \]

Answer 2

To solve the problem, we need to find the value of $k$ for the quadratic equation: 

$ x^2 + 4x + k = 0 $

given that the roots are real and equal.

1. Condition for Real and Equal Roots:
For a quadratic equation $ax^2 + bx + c = 0$, roots are real and equal if the discriminant $\Delta = b^2 - 4ac = 0$.

2. Calculate the Discriminant:
Here, $a = 1$, $b = 4$, and $c = k$.
$ \Delta = 4^2 - 4 \times 1 \times k = 16 - 4k $

3. Set Discriminant to Zero:
$16 - 4k = 0 \Rightarrow 4k = 16 \Rightarrow k = 4$

Final Answer:
The value of $k$ is $ {4} $.