The problem involves solving a quadratic equation using the quadratic formula. Based on the student's work, the quadratic equation being solved is $3x^2 - 13x + 10 = 0$. Step 1: Identify the coefficients of the quadratic equation. For $ax^2 + bx + c = 0$, we have $a=3$, $b=-13$, and $c=10$. Step 2: Apply the quadratic formula. The quadratic formula is given by: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Step 3: Substitute the values of $a$, $b$, and $c$ into the formula. $$ x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(3)(10)}}{2(3)} $$ Step 4: Simplify the expression. $$ x = \frac{13 \pm \sqrt{169 - 120}}{6} $$ Step 5: Calculate the value under the square root. $$ x = \frac{13 \pm \sqrt{49}}{6} $$ Step 6: Calculate the square root. $$ x = \frac{13 \pm 7}{6} $$ Step 7: Calculate the two possible solutions for $x$. $$ x_1 = \frac{13 + 7}{6} \quad \text{or} \quad x_2 = \frac{13 - 7}{6} $$ Step 8: Simplify the solutions. $$ x_1 = \frac{20}{6} \quad \text{or} \quad x_2 = \frac{6}{6} $$ $$ x_1 = \frac{10}{3} \quad \text{or} \quad x_2 = 1 $$ The final answers are: $$ x_1 = \boxed{\frac{10}{3}} \quad \text{or} \quad x_2 = \boxed{1} $$