How do you solve \[6{x^2} - 5x + 3 = 0\] using the quadratic formula?

Hint: Here in this question, we have to solve the given equation, the given equation is in the form of a quadratic equation. This is a quadratic equation for the variable x. By using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], we can determine the solutions.

Complete step-by-step solution:
The question involves the quadratic equation. To the quadratic equation we can find the roots by factoring or by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. consider the given equation \[6{x^2} - 5x + 3 = 0\].
In general, the quadratic equation is represented as \[a{x^2} + bx + c = 0\], when we compare the above equation to the general form of equation the values are as follows. a=6 b=-5 and c=3. Now substituting these values to the formula for obtaining the roots we have
\[\Rightarrow x = \dfrac{{ - ( - 5) \pm \sqrt {{{( - 5)}^2} - 4(6)(3)} }}{{2(6)}}\]
On simplifying the terms, we have
\[ \Rightarrow x = \dfrac{{5 \pm \sqrt {25 + 72} }}{{12}}\]
Now add 25 to 72 we get
\[ \Rightarrow x = \dfrac{{5 \pm \sqrt {97} }}{{12}}\]
The number 97 is not a perfect square number and we don’t have a square root for this. So, the square root of is carried out as it is so we have.
Therefore, we have \[x = \dfrac{{5 + \sqrt {97} }}{{12}}\] or \[x = \dfrac{{5 - \sqrt {97} }}{{12}}\]. We can simplify for further so we get
\[x = 1.237\] and \[x = - 0.4040\]
hence we have solved the quadratic equation and found the value of the variable x.
The equation is also solved by using the factorisation method.

Note: The quadratic equation can be solved by using the factorisation method and we also find the roots by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. While factorising we use sum product rule, the sum product rule is given as the product factors of the number c is equal to the sum of the factors which satisfies the value of b.