Question

The nature of roots of the equation ${{x}^{2}}+x+1=0$ is:
(A). Real and equal
(B). Real and unequal
(C). Imaginary and distinct
(D). Imaginary and equal

Answer 1

Hint: Find the discriminant of the equation given below and use the relation of discriminant with the nature of roots to reach to the answer. You might also need to use the property that imaginary roots appear in pairs for the polynomials with real coefficients.

Complete step-by-step solution -
For a quadratic equation of the form $a{{x}^{2}}+bx+c=0$ , the discriminant D is given by: $D={{b}^{2}}-4ac$ , and if discriminant is positive, the roots of the equation are real and distinct, while D=0 means real, and equal roots and D negative refers to imaginary roots.
Now using the above property for the equation ${{x}^{2}}+x+1=0$ . The discriminant of the equation is:
$D={{b}^{2}}-4ac=1-4\times 1\times 1=-3$
As discriminant is negative, we can say that the roots of the equation are imaginary. Also, the coefficients of the equation are real, and we know that the imaginary roots appear in conjugate pairs for polynomials with real coefficients, so the roots of the equation ${{x}^{2}}+x+1=0$ are imaginary and distinct.
Hence, the answer to the above question is option (c).

Note: While finding the roots of the quadratic equation, it is always prescribed to check the nature of the roots first, as this might help you to select the better way of finding the roots. Point to remember is that Middle term splitting is mostly useful when you have rational roots in case of imaginary or irrational roots; it becomes difficult to figure out the numbers manually.