Question

The discriminant of a quadratic equation is $-5$. Which answer describes the number and type of solutions of the equation
A) 1 and complex solution
B) 2 and real solutions
C) 2 and complex solutions
D) 1 and real solution

Answer 1

Hint: To solve this question we need basic knowledge of what a quadratic equation is and what it means to be a discriminant. How a discriminant is related with the type of solution or roots of the equation. There are three values for the discriminant that decides the type of roots. We will check them one by one.

Complete step by step answer:
Given that an equation of the form \[a{x^2} + bx + c = 0\] is said to be a quadratic equation.
The roots of this equation are found with the help of either factorization method or by using quadratic equation formula \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].
The term in the formula above \[\sqrt {{b^2} - 4ac} \] is called discriminant that decides the nature of the roots of the equation.
Now in the case above they have already proven that the discriminant is negative. Now no matter what the value is, then what will be the effect on the roots.
So if the discriminant is negative then there is no real root for that equation. Because we know that, \[\sqrt { - 1} = i\] turns into imaginary numbers or we can say complex numbers.
Thus it clears that, if the discriminant is less than zero that is negative then the roots are complex and since the equation is of degree two there will be two roots for that.
Thus option (C) is the correct option. That is 2 complex roots.

Note:
1. Note that, roots can be real and imaginary totally depending on the values of the discriminant. Now 1. if the value of \[\sqrt {{b^2} - 4ac} \] is greater than zero that is positive then the roots of the equation are real and are distinct, that is two different values for the variable exist.
2. and if the value of \[\sqrt {{b^2} - 4ac} \] is zero then the roots of the equation are real but they are the same.