How do you solve \[3{x^2} - 5x + 7 = 0\]?

Hint: Use method of determinant to solve for the value of x from the given quadratic equation. Compare the quadratic equation with general quadratic equation and substitute values in the formula of finding roots of the equation. Substitute the value of \[\sqrt { - 1} = i\] to convert the values under the square root into complex number
* For a general quadratic equation \[a{x^2} + bx + c = 0\], roots are given by formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Complete step-by-step answer:
We are given the quadratic equation \[3{x^2} - 5x + 7 = 0\] … (1)
We know that general quadratic equation is \[a{x^2} + bx + c = 0\]
On comparing with general quadratic equation \[a{x^2} + bx + c = 0\], we get \[a = 3,b = - 5,c = 7\]
Substitute the values of a, b and c in the formula of finding roots of the equation.
\[ \Rightarrow x = \dfrac{{ - ( - 5) \pm \sqrt {{{( - 5)}^2} - 4 \times 3 \times 7} }}{{2 \times 3}}\]
Square the values inside the square root in numerator of the fraction
\[ \Rightarrow x = \dfrac{{5 \pm \sqrt {25 - 84} }}{6}\]
Calculate the value under square root in the fraction
\[ \Rightarrow x = \dfrac{{5 \pm \sqrt { - 59} }}{6}\]
So, \[x = \dfrac{{5 + \sqrt { - 59} }}{6}\]and \[x = \dfrac{{5 - \sqrt { - 59} }}{6}\]
We know that value of \[\sqrt { - 1} = i\], we can break the answers under the square root and substitute the value of \[\sqrt { - 1} = i\]wherever required
So, \[x = \dfrac{{5 + \sqrt { - 1 \times 59} }}{6}\] and \[x = \dfrac{{5 - \sqrt { - 1 \times 59} }}{6}\]
We can break the product under the square root as product of square roots
So, \[x = \dfrac{{5 + \sqrt { - 1} \times \sqrt {59} }}{6}\] and \[x = \dfrac{{5 - \sqrt { - 1} \times \sqrt {59} }}{6}\]
Now we substitute the value of \[\sqrt { - 1} = i\] in both solutions
So, \[x = \dfrac{{5 + i\sqrt {59} }}{6}\]and \[x = \dfrac{{5 - i \times \sqrt {59} }}{6}\]

\[\therefore \]Solution of the equation \[3{x^2} - 5x + 7 = 0\] is \[x = \dfrac{{5 + i\sqrt {59} }}{6}\]and \[x = \dfrac{{5 - i \times \sqrt {59} }}{6}\]

Note:
Many students make the mistake of solving for the roots or values of x for this question using factorization method which is wrong. Many students leave their answer with a negative term under the square root which is wrong as we clearly know the value of the negative term under the square root means the presence of ‘i’ i.e. the number becomes a complex number.