If the roots of the equation x^2 - 4x + k = 0 are real and distinct, what is the

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Question: If the roots of the equation x^2 - 4x + k = 0 are real and distinct, what is the condition for k? (2023)

Options:

Correct Answer: k > 4

Exam Year: 2023

Solution:

The discriminant must be greater than zero for real and distinct roots: (-4)^2 - 4*1*k > 0, which simplifies to 16 - 4k > 0, or k < 4.

If the roots of the equation x^2 - 4x + k = 0 are real and distinct, what is the condition for k? (2023)

If the roots of the equation x^2 - 4x + k = 0 are real and distinct, what is the condition for k? (2023)

Step 1: Identify the equation given, which is x^2 - 4x + k = 0.
Step 2: Recognize that to find the roots of a quadratic equation, we use the discriminant formula, which is b^2 - 4ac.
Step 3: In our equation, a = 1, b = -4, and c = k.
Step 4: Substitute the values of a, b, and c into the discriminant formula: (-4)^2 - 4*1*k.
Step 5: Calculate (-4)^2, which equals 16, so we have 16 - 4k.
Step 6: For the roots to be real and distinct, the discriminant must be greater than zero: 16 - 4k > 0.
Step 7: Solve the inequality 16 - 4k > 0 by isolating k.
Step 8: Subtract 16 from both sides: -4k > -16.
Step 9: Divide both sides by -4 (remember to flip the inequality sign): k < 4.
Step 10: Conclude that the condition for k is k < 4.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; it must be greater than zero for the roots to be real and distinct.
Quadratic Equation – A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
Inequalities – Understanding how to manipulate and solve inequalities is crucial for determining the conditions on k.