Question: The equation x^2 - 4x + k = 0 has no real roots if k is:
Options:
< 4
≥ 4
≤ 4
> 4
Correct Answer: < 4
Solution:
The discriminant must be less than zero: (-4)^2 - 4*1*k < 0 leads to k > 4.
The equation x^2 - 4x + k = 0 has no real roots if k is:
Practice Questions
Q1
The equation x^2 - 4x + k = 0 has no real roots if k is:
< 4
≥ 4
≤ 4
> 4
Questions & Step-by-Step Solutions
The equation x^2 - 4x + k = 0 has no real roots if k is:
Step 1: Identify the equation given: x^2 - 4x + k = 0.
Step 2: Recognize that this is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 1, b = -4, and c = k.
Step 3: Understand that the discriminant (D) of a quadratic equation determines the nature of its roots. The formula for the discriminant is D = b^2 - 4ac.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = (-4)^2 - 4*1*k.
Step 5: Simplify the expression: D = 16 - 4k.
Step 6: For the equation to have no real roots, the discriminant must be less than zero: 16 - 4k < 0.
Step 7: Solve the inequality: 16 < 4k.
Step 8: Divide both sides by 4: 4 < k.
Step 9: Rewrite the result: k > 4.
Quadratic Equations – Understanding the conditions under which a quadratic equation has real or complex roots, specifically using the discriminant.
Discriminant – The discriminant of a quadratic equation determines the nature of the roots; it is calculated as b^2 - 4ac.
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