1. **State the problem:** We need to analyze and graph the quadratic equation $$b^2 - 4b + 4 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = -4$$, and $$c = 4$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0$$. 5. **Interpret the discriminant:** Since $$\Delta = 0$$, the quadratic has exactly one real root (a repeated root). 6. **Find the root:** Using the quadratic formula, $$b = \frac{-(-4) \pm \sqrt{0}}{2 \times 1} = \frac{4 \pm 0}{2} = 2$$. 7. **Factor the quadratic:** The equation can be factored as $$b^2 - 4b + 4 = (b - 2)^2 = 0$$. 8. **Graph interpretation:** The graph of $$y = b^2 - 4b + 4$$ is a parabola opening upwards with vertex at $$b = 2$$, $$y = 0$$, touching the x-axis at this point. **Final answer:** The quadratic equation $$b^2 - 4b + 4 = 0$$ has one real root at $$b = 2$$, and its graph is a parabola tangent to the x-axis at this point.