Let \( f(x)=3 x^{2}+x-2 \) be a quadratic function. (i) By completing the square, express \( f(x) \) in the form \( a(x+p)^{2} q \) where \( a, p \) and \( q \) are constants. \( x \)

To express the quadratic function \( f(x) = 3x^{2} + x - 2 \) in the form \( a(x + p)^{2} + q \) by completing the square, follow these steps: ### Step 1: Factor Out the Coefficient of \( x^2 \) Start by factoring out the coefficient of \( x^2 \) from the first two terms: \[ f(x) = 3x^{2} + x - 2 = 3\left(x^{2} + \frac{1}{3}x\right) - 2 \] ### Step 2: Complete the Square Inside the Parentheses To complete the square for the expression inside the parentheses \( \left(x^{2} + \frac{1}{3}x\right) \), follow these sub-steps: 1. **Find Half of the Coefficient of \( x \):** \[ \text{Half of } \frac{1}{3} = \frac{1}{6} \] 2. **Square This Value:** \[ \left(\frac{1}{6}\right)^2 = \frac{1}{36} \] 3. **Add and Subtract This Square Inside the Parentheses:** \[ x^{2} + \frac{1}{3}x = \left(x^{2} + \frac{1}{3}x + \frac{1}{36}\right) - \frac{1}{36} = \left(x + \frac{1}{6}\right)^2 - \frac{1}{36} \] ### Step 3: Substitute Back and Simplify Now, substitute the completed square back into the original expression: \[ f(x) = 3\left(\left(x + \frac{1}{6}\right)^2 - \frac{1}{36}\right) - 2 \] Distribute the 3 and combine like terms: \[ f(x) = 3\left(x + \frac{1}{6}\right)^2 - \frac{3}{36} - 2 = 3\left(x + \frac{1}{6}\right)^2 - \frac{1}{12} - 2 \] Convert the constant term to twelfths to combine with \(-\frac{1}{12}\): \[ -2 = -\frac{24}{12} \quad \Rightarrow \quad f(x) = 3\left(x + \frac{1}{6}\right)^2 - \frac{1}{12} - \frac{24}{12} = 3\left(x + \frac{1}{6}\right)^2 - \frac{25}{12} \] ### Final Form Thus, the quadratic function expressed by completing the square is: \[ f(x) = 3\left(x + \frac{1}{6}\right)^2 - \frac{25}{12} \] Here, the constants are: - \( a = 3 \) - \( p = \frac{1}{6} \) - \( q = -\frac{25}{12} \)