For the quadratic equation x² + 2x + k = 0 to have real roots, what is the condi
Practice Questions
Q1
For the quadratic equation x² + 2x + k = 0 to have real roots, what is the condition on k? (2021)
k ≥ 1
k ≤ 1
k > 1
k < 1
Questions & Step-by-Step Solutions
For the quadratic equation x² + 2x + k = 0 to have real roots, what is the condition on k? (2021)
Step 1: Identify the quadratic equation, which is x² + 2x + k = 0.
Step 2: Recall that for a quadratic equation ax² + bx + c = 0, the discriminant is given by the formula D = b² - 4ac.
Step 3: In our equation, a = 1, b = 2, and c = k.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = 2² - 4*1*k.
Step 5: Calculate 2², which is 4, so we have D = 4 - 4k.
Step 6: For the quadratic to have real roots, the discriminant must be non-negative: D ≥ 0.
Step 7: Set up the inequality: 4 - 4k ≥ 0.
Step 8: Rearrange the inequality to isolate k: 4 ≥ 4k.
Step 9: Divide both sides by 4: 1 ≥ k, or k ≤ 1.
Step 10: Conclude that for the quadratic equation to have real roots, k must be less than or equal to 1.
Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax² + bx + c = 0 is given by D = b² - 4ac. For the roots to be real, D must be non-negative (D ≥ 0).
Conditions for Real Roots – The condition for a quadratic equation to have real roots is that the discriminant must be greater than or equal to zero.
