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

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

Options:

Correct Answer: k > 1

Solution:

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

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

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

Step 1: Identify the equation given, which is x^2 - 2x + k = 0.
Step 2: Recognize that to find the roots of a quadratic equation, we use the discriminant.
Step 3: The discriminant formula for a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = -2, and c = k.
Step 5: Substitute the values of a, b, and c into the discriminant formula: D = (-2)^2 - 4*1*k.
Step 6: Calculate (-2)^2, which is 4, so we have D = 4 - 4k.
Step 7: For the roots to be real and distinct, the discriminant must be greater than 0: 4 - 4k > 0.
Step 8: Solve the inequality 4 - 4k > 0 by isolating k.
Step 9: Rearranging gives us 4 > 4k, or dividing both sides by 4 gives 1 > k.
Step 10: This means k must be less than 1 for the roots to be real and distinct.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the conditions for the nature of roots based on the discriminant.
Discriminant – The discriminant of a quadratic equation determines the nature of its roots: real and distinct, real and equal, or complex.