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What is the value of k for which the equation x^2 + kx + 9 = 0 has no real roots
What is the value of k for which the equation x^2 + kx + 9 = 0 has no real roots?
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Practice Questions
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Q1
What is the value of k for which the equation x^2 + kx + 9 = 0 has no real roots?
k < 6
k > 6
k = 6
k <= 6
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The discriminant must be negative: k^2 - 4*1*9 < 0 => k^2 < 36 => |k| < 6, hence k > 6.
Questions & Step-by-step Solutions
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Q
Q: What is the value of k for which the equation x^2 + kx + 9 = 0 has no real roots?
Solution:
The discriminant must be negative: k^2 - 4*1*9 < 0 => k^2 < 36 => |k| < 6, hence k > 6.
Steps: 9
Show Steps
Step 1: Identify the equation given, which is x^2 + kx + 9 = 0.
Step 2: Understand that for a quadratic equation to have no real roots, the discriminant must be less than zero.
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 = k, and c = 9. So, we calculate the discriminant: D = k^2 - 4*1*9.
Step 5: Simplify the discriminant: D = k^2 - 36.
Step 6: Set the discriminant less than zero for no real roots: k^2 - 36 < 0.
Step 7: Rearrange the inequality: k^2 < 36.
Step 8: Take the square root of both sides: |k| < 6.
Step 9: This means k can be between -6 and 6, so k must be greater than -6 and less than 6.
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