For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the

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Question: For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the minimum value of k? (2020)

Options:

Correct Answer: -1

Exam Year: 2020

Solution:

The discriminant must be non-negative: 2^2 - 4*1*k >= 0, thus k <= 1.

For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the minimum value of k? (2020)

For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the minimum value of k? (2020)

Step 1: Identify the quadratic equation, which is x^2 + 2x + k = 0.
Step 2: Recall that for a quadratic equation to have real roots, the discriminant must be non-negative.
Step 3: The discriminant (D) for the equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = 2, and c = k.
Step 5: Substitute the values into the discriminant formula: D = 2^2 - 4*1*k.
Step 6: Simplify the expression: D = 4 - 4k.
Step 7: Set the discriminant greater than or equal to zero for real roots: 4 - 4k >= 0.
Step 8: Rearrange the inequality: 4 >= 4k.
Step 9: Divide both sides by 4: 1 >= k.
Step 10: This means k must be less than or equal to 1 for the quadratic to have real roots.

Quadratic Equations – Understanding the conditions for real roots in quadratic equations, specifically using the discriminant.
Discriminant – The discriminant (b^2 - 4ac) determines the nature of the roots of a quadratic equation.