Question

The roots of the equation $2{x^2} + 3x + 2 = 0$are
A. Real, rational and equal
B. Real, rational and unequal
C. Real, irrational and unequal
D. Non real (imaginary)

Answer 1

Hint: We can find the discriminant of the equation using the formula $D = {b^2} - 4ac$. If the discriminant of the equation is zero, it will have equal real roots. If the discriminant is less than zero, the equation will have imaginary roots. If the discriminant is greater than zero, the equation will have 2 real and distinct roots. If the square root of the discriminant is rational, the roots will be rational.

Complete step by step answer:

We have the quadratic equation, $2{x^2} + 3x + 2 = 0$.
For a quadratic equation, nature of roots can be determined by taking the discriminant of the equation.
Discriminant of a quadratic equation of the form $a{x^2} + bx + c = 0$ is given by $D = {b^2} - 4ac$
Therefore, the discriminant of the equation $2{x^2} + 3x + 2 = 0$ is
$D = {2^2} - 4 \times 2 \times 2$
On simplification we get,
$ \Rightarrow D = 4 - 16$
$ \Rightarrow D = - 12$
As the discriminant is negative, the given equation will have imaginary roots.
So, the correct answer is option D.

Note: After finding the discriminant, the roots of the equation can be calculated using the quadratic formula. For an equation of the form $a{x^2} + bx + c = 0$, its roots are given by the formula,
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. Here the term inside the square root is the discriminant. While taking the discriminant or applying quadratic formula we must make sure that the equation is in the standard form of $a{x^2} + bx + c = 0$. We proceed our calculation to find the imaginary roots by introducing an imaginary number, $i = \sqrt { - 1} $.