How do you solve using the quadratic formula?

Hint: A polynomial of degree two is called a quadratic equation and the values of the unknown variable for which the value of the function comes out to be zero are called the zeros/factors/solutions of the polynomial equation. Some methods for finding the roots of a quadratic equation are factorization, completing the square, graphs, quadratic formula, etc.

Complete step by step solution:
We have to solve the quadratic equation given in the question using the quadratic formula, so for that, we will rewrite the equation and compare it to the standard equation form and then we will plug in the values of the coefficients in the quadratic formula to get the correct answer.
The equation given is $3{x^2} + 5x + 1 = 0$
On comparing the given equation with the standard quadratic equation $a{x^2} + bx + c = 0$, we get –
$a = 3,\,b = 5,\,c = 1$
The Quadratic formula is given as –
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Putting the values in the above equation, we get –
$
\Rightarrow x = \dfrac{{ - 5 \pm \sqrt {{{(5)}^2} - 4(3)(1)} }}{{2(3)}} \\
\Rightarrow x = \dfrac{{ - 5 \pm \sqrt {25 - 12} }}{6} \\
\Rightarrow x = \dfrac{{ - 5 \pm \sqrt {13} }}{6} \\
 $
Hence the zeros of the given equation are $\dfrac{{ - 5 + \sqrt {13} }}{6}$ and $\dfrac{{ - 5 - \sqrt {13} }}{6}$.

Note: An expression containing numerical values along with alphabets is called an algebraic expression; a polynomial equation is obtained when the alphabets representing an unknown variable quantity are raised to some power such that the exponent is a non-negative integer. The highest exponent of the polynomial is known as the degree of the polynomial equation. The value of y is zero on the x-axis, so the roots of an equation are simply the x-intercepts. Usually, when we fail to find the factors of the equation like in this question, the factors cannot be made, we use the quadratic formula.