Find the value of k for which the equation x² + 4x + k = 0 has no real roots. (2

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Question: Find the value of k for which the equation x² + 4x + k = 0 has no real roots. (2020)

Options:

Correct Answer: -6

Solution:

The discriminant must be negative: 4² - 4*1*k < 0, which gives k > 4, so the minimum value is -6.

Find the value of k for which the equation x² + 4x + k = 0 has no real roots. (2020)

Find the value of k for which the equation x² + 4x + k = 0 has no real roots. (2020)

Step 1: Identify the equation given, which is x² + 4x + k = 0.
Step 2: Recognize that for a quadratic equation to have no real roots, the discriminant must be negative.
Step 3: Write down the formula for the discriminant, which is D = b² - 4ac. Here, a = 1, b = 4, and c = k.
Step 4: Substitute the values into the discriminant formula: D = 4² - 4*1*k.
Step 5: Simplify the expression: D = 16 - 4k.
Step 6: Set the discriminant less than zero for no real roots: 16 - 4k < 0.
Step 7: Solve the inequality: 16 < 4k.
Step 8: Divide both sides by 4: 4 < k.
Step 9: This means k must be greater than 4 for the equation to have no real roots.
Step 10: The minimum value of k that satisfies this condition is just above 4.

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 = b² - 4ac.
Inequalities – Solving inequalities to find the range of values for k that satisfy the condition of having no real roots.