Question: If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condition on k?
Options:
k < 0
k > 0
k >= 0
k <= 0
Correct Answer: k < 0
Solution:
For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0, hence k > 1.
If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condit
Practice Questions
Q1
If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condition on k?
k < 0
k > 0
k >= 0
k <= 0
Questions & Step-by-Step Solutions
If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condition on k?
Correct Answer: k > 1
Step 1: Identify the quadratic equation, which is x^2 + 2x + 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 = 2, and c = k.
Step 5: Substitute the values into the discriminant formula: D = 2^2 - 4*1*k.
Step 6: Simplify the expression: D = 4 - 4k.
Step 7: Set the condition for no real roots: 4 - 4k < 0.
Step 8: Solve the inequality: 4 < 4k.
Step 9: Divide both sides by 4: 1 < k.
Step 10: Rewrite the condition: k > 1.
Quadratic Equations – Understanding the properties of quadratic equations, particularly the role of the discriminant in determining the nature of the roots.
Discriminant – The discriminant (b^2 - 4ac) helps to determine whether a quadratic equation has real roots, complex roots, or repeated roots.
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