1. **State the problem:** Given the quadratic function $g(x) = 3x^2 - 8x + 2$, (i) write $g(x)$ in the form $a(x+h)^2 + k$ (vertex form). (ii) solve the equation $g(x) = 0$ and write the solutions correct to 2 decimal places. 2. **Formula and rules:** To write a quadratic in vertex form, use completing the square: $$g(x) = a(x+h)^2 + k$$ where $a$ is the coefficient of $x^2$, and $h,k$ are constants. 3. **Step (i) Completing the square:** Start with $$g(x) = 3x^2 - 8x + 2$$ Factor out 3 from the first two terms: $$g(x) = 3\left(x^2 - \frac{8}{3}x\right) + 2$$ Complete the square inside the parentheses: Take half of $-\frac{8}{3}$, which is $-\frac{4}{3}$, then square it: $$\left(-\frac{4}{3}\right)^2 = \frac{16}{9}$$ Add and subtract $\frac{16}{9}$ inside the parentheses: $$g(x) = 3\left(x^2 - \frac{8}{3}x + \frac{16}{9} - \frac{16}{9}\right) + 2$$ Group the perfect square trinomial: $$g(x) = 3\left(\left(x - \frac{4}{3}\right)^2 - \frac{16}{9}\right) + 2$$ Distribute 3: $$g(x) = 3\left(x - \frac{4}{3}\right)^2 - 3 \times \frac{16}{9} + 2$$ Simplify: $$g(x) = 3\left(x - \frac{4}{3}\right)^2 - \frac{48}{9} + 2 = 3\left(x - \frac{4}{3}\right)^2 - \frac{16}{3} + 2$$ Convert 2 to thirds: $$2 = \frac{6}{3}$$ So, $$g(x) = 3\left(x - \frac{4}{3}\right)^2 - \frac{16}{3} + \frac{6}{3} = 3\left(x - \frac{4}{3}\right)^2 - \frac{10}{3}$$ 4. **Answer for (i):** $$g(x) = 3\left(x - \frac{4}{3}\right)^2 - \frac{10}{3}$$ 5. **Step (ii) Solve $g(x) = 0$:** Set the vertex form equal to zero: $$3\left(x - \frac{4}{3}\right)^2 - \frac{10}{3} = 0$$ Add $\frac{10}{3}$ to both sides: $$3\left(x - \frac{4}{3}\right)^2 = \frac{10}{3}$$ Divide both sides by 3: $$\cancel{3}\left(x - \frac{4}{3}\right)^2 = \frac{10}{3} \div \cancel{3} = \frac{10}{9}$$ Take the square root of both sides: $$x - \frac{4}{3} = \pm \sqrt{\frac{10}{9}} = \pm \frac{\sqrt{10}}{3}$$ Add $\frac{4}{3}$ to both sides: $$x = \frac{4}{3} \pm \frac{\sqrt{10}}{3} = \frac{4 \pm \sqrt{10}}{3}$$ 6. **Calculate decimal approximations:** $$\sqrt{10} \approx 3.1623$$ So, $$x_1 = \frac{4 + 3.1623}{3} = \frac{7.1623}{3} \approx 2.39$$ $$x_2 = \frac{4 - 3.1623}{3} = \frac{0.8377}{3} \approx 0.28$$ **Final answers:** (i) $$g(x) = 3\left(x - \frac{4}{3}\right)^2 - \frac{10}{3}$$ (ii) $$x \approx 2.39 \text{ or } x \approx 0.28$$