The quadratic equation $ax^{2}+bx+c=0$ will always have imaginary roots if:
- $ a<-1,0< c <1,b<0 $
- $a<-1,-1< c < 0,0 < b < 1$
- $a<-1,c<0,b>1 $
- $ a <-1,c < -1,1< b < 2 $
Reveal answer
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D
Correct answer
Explanation
Roots are imaginary if b^2 - 4ac < 0. Testing option D: a < -1, c < -1, 1 < b < 2. Let a = -2, c = -2, b = 1.5. b^2 - 4ac = 2.25 - 4(-2)(-2) = 2.25 - 16 = -13.75 < 0. This works.
AI explanation
To guarantee a quadratic equation has imaginary roots, its discriminant D = b^2 - 4ac must be strictly less than zero, meaning b^2 < 4ac. In the fourth option, a is less than -1 and c is less than -1, so their product ac is positive and greater than 1. Multiplying by 4 means 4ac is substantially positive, and since b is between 1 and 2, b^2 is less than 4. Consequently, b^2 is always smaller than 4ac, fulfilling the requirement for imaginary roots.