For the quadratic equation x^2 + 4x + k = 0 to have real roots, what is the cond
Practice Questions
Q1
For the quadratic equation x^2 + 4x + k = 0 to have real roots, what is the condition on k?
k >= 4
k <= 4
k > 0
k < 0
Questions & Step-by-Step Solutions
For the quadratic equation x^2 + 4x + k = 0 to have real roots, what is the condition on k?
Correct Answer: k <= 4
Step 1: Identify the quadratic equation, which is x^2 + 4x + k = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the discriminant is given by the formula D = b^2 - 4ac.
Step 3: In our equation, a = 1, b = 4, and c = k.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = 4^2 - 4*1*k.
Step 5: Calculate 4^2, which is 16, so we have D = 16 - 4k.
Step 6: For the quadratic equation to have real roots, the discriminant must be non-negative: D >= 0.
Step 7: Set up the inequality: 16 - 4k >= 0.
Step 8: Rearrange the inequality to isolate k: 16 >= 4k.
Step 9: Divide both sides of the inequality by 4: 4 >= k.
Step 10: Rewrite the inequality: k <= 4.
Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. For the roots to be real, D must be non-negative (D >= 0).
Conditions for Real Roots – Understanding that for a quadratic equation to have real roots, the discriminant must be greater than or equal to zero.
