The equation x^2 - 2x + k = 0 has roots that are both positive. What is the range of k?
Practice Questions
1 question
Q1
The equation x^2 - 2x + k = 0 has roots that are both positive. What is the range of k?
k < 0
k > 0
k > 1
k < 1
For both roots to be positive, k must be greater than 1 (from Vieta's formulas).
Questions & Step-by-step Solutions
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Q
Q: The equation x^2 - 2x + k = 0 has roots that are both positive. What is the range of k?
Solution: For both roots to be positive, k must be greater than 1 (from Vieta's formulas).
Steps: 12
Step 1: Understand that the equation x^2 - 2x + k = 0 is a quadratic equation.
Step 2: Recall that a quadratic equation can have two roots, and we want both roots to be positive.
Step 3: Use Vieta's formulas, which tell us that the sum of the roots (let's call them r1 and r2) is equal to 2 (the coefficient of x with the opposite sign).
Step 4: Since both roots are positive, their sum (r1 + r2) must be greater than 0, which is already satisfied because it equals 2.
Step 5: Now, Vieta's formulas also tell us that the product of the roots (r1 * r2) is equal to k.
Step 6: For both roots to be positive, the product r1 * r2 must also be greater than 0, which means k must be greater than 0.
Step 7: However, we also need to ensure that the roots are not only positive but also real. This requires the discriminant (b^2 - 4ac) to be non-negative.
Step 8: Calculate the discriminant for our equation: (-2)^2 - 4(1)(k) = 4 - 4k.
Step 9: Set the discriminant greater than or equal to 0: 4 - 4k >= 0.
Step 10: Solve for k: 4 >= 4k, which simplifies to k <= 1.
Step 11: Combine the conditions: k must be greater than 0 (for positive roots) and less than or equal to 1 (for real roots).
