Find the value of k for which the equation x² + kx + 9 = 0 has no real roots. (2

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Question: Find the value of k for which the equation x² + kx + 9 = 0 has no real roots. (2023)

Options:

Correct Answer: -6

Exam Year: 2023

Solution:

For no real roots, the discriminant must be negative: k² - 4*1*9 < 0, thus k² < 36, hence k < -6 or k > 6.

Find the value of k for which the equation x² + kx + 9 = 0 has no real roots. (2023)

Find the value of k for which the equation x² + kx + 9 = 0 has no real roots. (2023)

Step 1: Identify the equation given, which is x² + kx + 9 = 0.
Step 2: Understand that for a quadratic equation to have no real roots, the discriminant must be negative.
Step 3: Recall the formula for the discriminant, which is D = b² - 4ac. In our equation, a = 1, b = k, and c = 9.
Step 4: Substitute the values into the discriminant formula: D = k² - 4*1*9.
Step 5: Simplify the expression: D = k² - 36.
Step 6: Set the discriminant less than zero for no real roots: k² - 36 < 0.
Step 7: Rearrange the inequality: k² < 36.
Step 8: Take the square root of both sides: |k| < 6.
Step 9: This means k must be between -6 and 6: -6 < k < 6.
Step 10: Therefore, the values of k for which the equation has no real roots are k < -6 or k > 6.

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 no real roots.