For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the
Practice Questions
Q1
For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the condition on k? (2023)
k < 1
k > 1
k >= 1
k <= 1
Questions & Step-by-Step Solutions
For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the condition on k? (2023)
Step 1: Identify the quadratic equation, which is x^2 + 2x + 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 = 2, and c = k.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = 2^2 - 4*1*k.
Step 5: Simplify the expression: D = 4 - 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: 4 - 4k >= 0.
Step 8: Rearrange the inequality to isolate k: 4 >= 4k.
Step 9: Divide both sides by 4: 1 >= k.
Step 10: This means k must be less than or equal to 1: k <= 1.
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 the condition under which a quadratic equation has real roots, specifically focusing on the value of k in relation to the discriminant.
