Question

The roots of the equation \[3{{x}^{2}}-4x+3=0\] are :
A.Real and unequal
B.Real and equal
C.Imaginary
D.None of these

Answer 1

Hint: -In this question, we will use the discriminant of a quadratic equation which is used to qualitatively check the roots of the quadratic equation whether the roots are real and distinct, real and equal or imaginary.

Complete step-by-step answer:
The most important formula that will be used in solving this question is as follows
So, the formula to calculate the discriminant of a quadratic equation is as follows
\[D=\sqrt{{{b}^{2}}-4ac}\]
(This is the discriminant of the quadratic equation \[a{{x}^{2}}+bx+c\])
 If D>0, then the quadratic equation has real and distinct roots.
If D=0, then the quadratic equation has real and equal roots.
If D<0, then the quadratic equation has imaginary roots.
As mentioned in the question, we have to check whether the roots of the given quadratic equation are real and distinct, real and equal or imaginary.
Now, using the formula that is given in the hint, we can calculate the discriminant of the given quadratic equation as follows
\[\begin{align}
  & 3{{x}^{2}}-4x+3=0 \\
 & \left[ D=\sqrt{{{b}^{2}}-4ac} \right] \\
 & \Rightarrow D=\sqrt{{{(-4)}^{2}}-4\times 3\times 3} \\
 & \Rightarrow D=\sqrt{16-36}=\sqrt{-20} \\
\end{align}\]
Now, as D<0, hence, the roots of the given quadratic equation are imaginary.

Note:-It is very important to know the principle of discriminant of a quadratic equation and also, that
If D>0, then the quadratic equation has real and distinct roots.
If D=0, then the quadratic equation has real and equal roots.
If D<0, then the quadratic equation has imaginary roots.
Because without knowing these relations, one can never get to the correct solution.