1. **Problem:** Identify the center and radius of the circle given by the equation $$(x - 4)^2 + (y - 1)^2 = 3$$ 2. **Formula:** The standard form of a circle's equation is $$(x - h)^2 + (y - k)^2 = r^2$$ where $(h,k)$ is the center and $r$ is the radius. 3. **Identify center and radius:** Comparing, center is $(4,1)$ and radius is $$\sqrt{3}$$. 1. **Problem:** Identify the center and radius of the circle given by $$x^2 + y^2 + 2x - 6y + 16 = 0$$ 2. **Formula:** Complete the square for $x$ and $y$ terms to rewrite in standard form. 3. **Complete the square:** $$x^2 + 2x + y^2 - 6y = -16$$ Add and subtract inside to complete squares: $$x^2 + 2x + 1 - 1 + y^2 - 6y + 9 - 9 = -16$$ Rewrite: $$(x + 1)^2 - 1 + (y - 3)^2 - 9 = -16$$ Simplify: $$(x + 1)^2 + (y - 3)^2 - 10 = -16$$ Add 10 to both sides: $$(x + 1)^2 + (y - 3)^2 = -16 + 10$$ $$(x + 1)^2 + (y - 3)^2 = -6$$ Since radius squared is negative, no real circle exists. 1. **Problem:** Factor completely $$7x^2 - 8x - 12$$ 2. **Method:** Use factoring by grouping or quadratic formula to find roots. 3. **Calculate discriminant:** $$D = (-8)^2 - 4 \times 7 \times (-12) = 64 + 336 = 400$$ 4. **Roots:** $$x = \frac{8 \pm \sqrt{400}}{2 \times 7} = \frac{8 \pm 20}{14}$$ 5. **Roots values:** $$x_1 = \frac{8 + 20}{14} = 2$$ $$x_2 = \frac{8 - 20}{14} = -\frac{6}{7}$$ 6. **Factor form:** $$7x^2 - 8x - 12 = 7(x - 2)(x + \frac{6}{7}) = (7x + 6)(x - 2)$$ 1. **Problem:** Factor completely $$3n^2 + 25n - 18$$ 2. **Calculate discriminant:** $$D = 25^2 - 4 \times 3 \times (-18) = 625 + 216 = 841$$ 3. **Roots:** $$n = \frac{-25 \pm \sqrt{841}}{2 \times 3} = \frac{-25 \pm 29}{6}$$ 4. **Roots values:** $$n_1 = \frac{-25 + 29}{6} = \frac{4}{6} = \frac{2}{3}$$ $$n_2 = \frac{-25 - 29}{6} = \frac{-54}{6} = -9$$ 5. **Factor form:** $$3n^2 + 25n - 18 = 3(n - \frac{2}{3})(n + 9) = (3n - 2)(n + 9)$$ 1. **Problem:** Sketch the graph of $$f(x) = -(x - 2)^2 + 2$$ 2. **Form:** This is a parabola in vertex form $$f(x) = a(x - h)^2 + k$$ with vertex at $(h,k)$. 3. **Identify vertex and direction:** Vertex is $(2,2)$, and since $a = -1 < 0$, parabola opens downward. 1. **Problem:** Sketch the graph of $$f(x) = -(x + 2)^2 + 3$$ 2. **Form:** Vertex form with vertex at $(-2,3)$ and opens downward. **Summary:** - Problem count: 6 - Only first problem fully solved: circle center and radius for $$(x - 4)^2 + (y - 1)^2 = 3$$ Final answer for first problem: Center: $(4,1)$ Radius: $$\sqrt{3}$$