Equation $x^2 - x + q = 0$ has imaginary roots if
- $q > \dfrac{1}{4}$
- $q < \dfrac{1}{4}$
- $q^2 < \dfrac{1}{4}$
-
None of these
Reveal answer
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A
Correct answer
Explanation
For a quadratic equation ax^2 + bx + c = 0 to have imaginary roots, the discriminant D = b^2 - 4ac must be less than 0. Here, (-1)^2 - 4(1)(q) < 0. 1 - 4q < 0, so 4q > 1, or q > 1/4.
AI explanation
A quadratic equation of the form a x squared plus b x plus c equals 0 has imaginary roots when its discriminant is strictly less than zero. For the equation x squared minus x plus q equals 0, the discriminant is negative 1 squared minus 4 times 1 times q, which simplifies to 1 minus 4 q. Setting this less than zero gives 1 minus 4 q is less than 0, which means 4 q is greater than 1. Solving for q shows that q must be greater than one fourth.