The roots of the equation x^2 - 2x + k = 0 are real and distinct if k is:

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Question: The roots of the equation x^2 - 2x + k = 0 are real and distinct if k is:

Options:

Correct Answer: < 1

Solution:

The discriminant must be greater than zero: (-2)^2 - 4*1*k > 0, which simplifies to k < 1.

The roots of the equation x^2 - 2x + k = 0 are real and distinct if k is:

The roots of the equation x^2 - 2x + k = 0 are real and distinct if k is:

Step 1: Identify the equation given, which is x^2 - 2x + k = 0.
Step 2: Recognize that for the roots of a quadratic equation to be real and distinct, the discriminant must be greater than zero.
Step 3: Write down the formula for the discriminant, which is b^2 - 4ac. Here, a = 1, b = -2, and c = k.
Step 4: Substitute the values into the discriminant formula: (-2)^2 - 4*1*k.
Step 5: Simplify the expression: 4 - 4k.
Step 6: Set up the inequality for the discriminant to be greater than zero: 4 - 4k > 0.
Step 7: Solve the inequality: 4 > 4k, which simplifies to 1 > k or k < 1.
Step 8: Conclude that the roots of the equation are real and distinct if k is less than 1.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the conditions for real and distinct roots.
Discriminant – Using the discriminant (b^2 - 4ac) to determine the nature of the roots of a quadratic equation.