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The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition
The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition on k? (2020)
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The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition on k? (2020)
k < 9
k > 9
k = 9
k ≤ 9
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For no real roots, the discriminant must be less than zero: 6^2 - 4*1*k < 0, which gives k > 9.
Questions & Step-by-step Solutions
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Q
Q: The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition on k? (2020)
Solution:
For no real roots, the discriminant must be less than zero: 6^2 - 4*1*k < 0, which gives k > 9.
Steps: 10
Show Steps
Step 1: Identify the quadratic equation, which is x^2 + 6x + k = 0.
Step 2: Recall that a quadratic equation has no real roots if its discriminant is 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 = 6, and c = k.
Step 5: Substitute the values into the discriminant formula: D = 6^2 - 4*1*k.
Step 6: Simplify the expression: D = 36 - 4k.
Step 7: Set the condition for no real roots: 36 - 4k < 0.
Step 8: Solve the inequality: 36 < 4k.
Step 9: Divide both sides by 4: 9 < k.
Step 10: Rewrite the condition: k > 9.
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