For the quadratic equation x^2 + 2x + k = 0 to have no real roots, k must be:

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Question: For the quadratic equation x^2 + 2x + k = 0 to have no real roots, k must be:

Options:

Correct Answer: < 0

Solution:

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

For the quadratic equation x^2 + 2x + k = 0 to have no real roots, k must be:

For the quadratic equation x^2 + 2x + k = 0 to have no real roots, k must be:

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: Calculate 2^2, which is 4. So, D = 4 - 4k.
Step 6: For the quadratic to have no real roots, the discriminant must be less than 0: 4 - 4k < 0.
Step 7: Rearrange the inequality: 4 < 4k.
Step 8: Divide both sides of the inequality by 4: 1 < k.
Step 9: This means k must be greater than 1.

Quadratic Equations – Understanding the conditions under which a quadratic equation has real or complex roots, specifically using the discriminant.
Discriminant – The formula used to determine the nature of the roots of a quadratic equation, given by D = b^2 - 4ac.
Inequalities – Solving inequalities to find the range of values for k that satisfy the condition of no real roots.