The roots of the quadratic equation

are ( \frac{1}{3} ) and ( 1 ).

To determine the roots of a quadratic equation in the standard form ( ax^2 + bx + c = 0 ), we can utilize the quadratic formula:

In our equation, we have ( a = 3 ), ( b = -4 ), and ( c = 1 ). Substituting these values into the formula gives us:

Calculating step-by-step:

1. Simplify the expression:

2. Continue simplifying under the square root:

3. Calculate the square root:

This results in two possible values for ( x ):

1. When we take the positive root:

2. When we take the negative root:

Thus, the roots of the equation are ( \frac{1}{3} ) and ( 1 ).

To verify these roots, we can substitute them back into the original equation:

1. Checking ( x = \frac{1}{3} ):

Simplifying this gives:

This simplifies to:

2. Checking ( x = 1 ):

This simplifies to:

Which is:

Both roots satisfy the original quadratic equation, confirming that our calculations are correct.