1. **State the problem:** We need to draw the graph of the quadratic function $$y = x^2 - 4x + 3$$ using a scale of 1 cm = 1 unit on both axes, and then use the graph to solve the equation $$x^2 - 4x + 3 = 0$$. 2. **Recall the quadratic function form:** The general form is $$y = ax^2 + bx + c$$ where $$a=1$$, $$b=-4$$, and $$c=3$$. 3. **Find the roots (x-intercepts) of the quadratic:** Solve $$x^2 - 4x + 3 = 0$$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 5. **Calculate the discriminant:** $$b^2 - 4ac = (-4)^2 - 4 \times 1 \times 3 = 16 - 12 = 4$$. 6. **Calculate the roots:** $$x = \frac{-(-4) \pm \sqrt{4}}{2 \times 1} = \frac{4 \pm 2}{2}$$ 7. **Evaluate each root:** - $$x = \frac{4 + 2}{2} = \frac{6}{2} = 3$$ - $$x = \frac{4 - 2}{2} = \frac{2}{2} = 1$$ 8. **Find the vertex:** The vertex $$x$$-coordinate is $$x = -\frac{b}{2a} = -\frac{-4}{2 \times 1} = 2$$. 9. **Calculate the vertex y-coordinate:** $$y = (2)^2 - 4 \times 2 + 3 = 4 - 8 + 3 = -1$$. 10. **Plot points:** - Vertex at $$(2, -1)$$ - Roots at $$(1, 0)$$ and $$(3, 0)$$ - Additional points for accuracy, e.g., $$x=0$$ gives $$y=3$$. 11. **Draw the parabola:** Using the points and scale, sketch the curve opening upwards. 12. **Use the graph to solve the equation:** The solutions to $$x^2 - 4x + 3 = 0$$ are the x-values where the graph crosses the x-axis, which are $$x=1$$ and $$x=3$$. **Final answer:** The solutions to the equation are $$\boxed{1 \text{ and } 3}$$.