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For which value of k does the quadratic equation x^2 - kx + 4 = 0 have no real r
For which value of k does the quadratic equation x^2 - kx + 4 = 0 have no real roots?
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Practice Questions
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Q1
For which value of k does the quadratic equation x^2 - kx + 4 = 0 have no real roots?
k < 4
k = 4
k > 4
k ≤ 4
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The discriminant must be negative: k^2 - 16 < 0, hence k > 4.
Questions & Step-by-step Solutions
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Q
Q: For which value of k does the quadratic equation x^2 - kx + 4 = 0 have no real roots?
Solution:
The discriminant must be negative: k^2 - 16 < 0, hence k > 4.
Steps: 10
Show Steps
Step 1: Identify the quadratic equation, which is x^2 - kx + 4 = 0.
Step 2: Recall that a quadratic equation has no real roots when its discriminant is less than zero.
Step 3: The discriminant (D) for the equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = -k, and c = 4. So, the discriminant is D = (-k)^2 - 4(1)(4).
Step 5: Simplify the discriminant: D = k^2 - 16.
Step 6: Set the discriminant less than zero for no real roots: k^2 - 16 < 0.
Step 7: Solve the inequality: k^2 < 16.
Step 8: Take the square root of both sides: |k| < 4.
Step 9: This means k is between -4 and 4: -4 < k < 4.
Step 10: However, we are interested in the values of k that make the discriminant negative, which means k must be greater than 4: k > 4.
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