Final answer:
To solve the quadratic function, set it to zero and apply the quadratic formula with the coefficients. Multiply through to remove any fractions, then proceed with the formula to find the values of x.
Step-by-step explanation:
To solve the quadratic equation f(x) = \(\frac{1}{3}x^2 - 2x + 8\), we need to set it to zero and use the quadratic formula, which applies to equations of the form ax² + bx + c = 0. The quadratic formula is x = \(-\frac{b}{2a} \pm \sqrt{(\frac{b}{2a})^2 - \frac{c}{a}}\). In this case, our equation is not yet set to zero, so we let f(x) = 0 to obtain \(\frac{1}{3}x^2 - 2x + 8 = 0\). Multiplying through by 3 to remove the fraction, we get x² - 6x + 24 = 0. Now, apply the quadratic formula with a = 1, b = -6, and c = 24. After solving, we find the values of x that satisfy the equation.