Question

If the equation \(x^2 + 4x + k = 0\) has real and distinct roots then :

• A. \(k<4\)
• B. \(k>4\)
• C. \(k \geq 4\)
• D. \(k \leq 4\)

Hint:

For a quadratic equation \(ax^2 + bx + c = 0\) to have {real and distinct roots}, the discriminant \(D = b^2 - 4ac\) must be strictly positive (\(D>0\)). 1. Given equation: \(x^2 + 4x + k = 0\). So, \(a=1, b=4, c=k\). 2. Set up the discriminant inequality: \(b^2 - 4ac>0\). \((4)^2 - 4(1)(k)>0\) \(16 - 4k>0\) 3. Solve for \(k\): \(16>4k\) \(4>k\), which means \(k<4\).

Answer 1

The Correct Option is A

Concept: For a quadratic equation \(ax^2 + bx + c = 0\), the nature of its roots is determined by the discriminant, \(D = b^2 - 4ac\).
If \(D>0\), the roots are real and distinct (unequal).
If \(D = 0\), the roots are real and equal (repeated).
If \(D<0\), the roots are complex (not real). The question states that the equation has "real and distinct roots". Step 1: Identify coefficients a, b, and c from the given equation The given quadratic equation is \(x^2 + 4x + k = 0\). Comparing with \(ax^2 + bx + c = 0\):
\(a = 1\)
\(b = 4\)
\(c = k\) Step 2: Apply the condition for real and distinct roots For real and distinct roots, the discriminant \(D\) must be greater than zero: \[ D>0 \] \[ b^2 - 4ac>0 \] Step 3: Substitute the coefficients into the discriminant inequality \[ (4)^2 - 4(1)(k)>0 \] \[ 16 - 4k>0 \] Step 4: Solve the inequality for k \[ 16>4k \] Divide by 4 (which is a positive number, so the inequality sign does not change): \[ \frac{16}{4}>k \] \[ 4>k \] This can also be written as: \[ k<4 \] The condition for the equation to have real and distinct roots is \(k<4\). This matches option (1).