1. **Problem:** Solve and analyze the quadratic equation $x^2 - 25 = 0$. 2. **Formula and rules:** The quadratic equation is $ax^2 + bx + c = 0$. The axis of symmetry is $x = -\frac{b}{2a}$, the vertex is at $(x, y)$ where $y = f(x)$, the y-intercept is $c$, and the x-intercepts are solutions to the equation. 3. **Solution:** $$x^2 - 25 = 0 \Rightarrow x^2 = 25 \Rightarrow x = \pm 5$$ 4. **Axis of symmetry:** $$x = -\frac{0}{2 \times 1} = 0$$ 5. **Vertex:** $$y = (0)^2 - 25 = -25$$ Vertex = (0, -25)$$ 6. **Y-intercept:** $$y = -25$$ 7. **X-intercepts:** $$x = 5, -5$$ --- 1. **Problem:** Solve and analyze $-x^2 - 4x = 0$. 2. **Rewrite:** $$-x^2 - 4x = 0 \Rightarrow x^2 + 4x = 0$$ 3. **Factor:** $$x(x + 4) = 0 \Rightarrow x = 0 \text{ or } x = -4$$ 4. **Axis of symmetry:** $$x = -\frac{4}{2 \times 1} = -2$$ 5. **Vertex:** $$y = -( -2)^2 - 4(-2) = -4 + 8 = 4$$ Vertex = (-2, 4)$$ 6. **Y-intercept:** $$y = 0$$ 7. **X-intercepts:** $$x = 0, -4$$ --- 1. **Problem:** Solve and analyze $-2x^2 + 5x + 6 = 0$. 2. **Use quadratic formula:** $$a = -2, b = 5, c = 6$$ $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-5 \pm \sqrt{25 - 4(-2)(6)}}{2(-2)} = \frac{-5 \pm \sqrt{25 + 48}}{-4} = \frac{-5 \pm \sqrt{73}}{-4}$$ 3. **Approximate roots:** $$x_1 = \frac{-5 + 8.544}{-4} = \frac{3.544}{-4} = -0.886$$ $$x_2 = \frac{-5 - 8.544}{-4} = \frac{-13.544}{-4} = 3.386$$ 4. **Axis of symmetry:** $$x = -\frac{5}{2(-2)} = \frac{5}{4} = 1.25$$ 5. **Vertex:** $$y = -2(1.25)^2 + 5(1.25) + 6 = -2(1.5625) + 6.25 + 6 = -3.125 + 12.25 = 9.125$$ Vertex = (1.25, 9.125)$$ 6. **Y-intercept:** $$y = 6$$ 7. **X-intercepts:** $$x \approx -0.886, 3.386$$ --- 1. **Problem:** Solve and analyze $x^2 - 10x + 25 = 0$. 2. **Recognize perfect square:** $$x^2 - 10x + 25 = (x - 5)^2 = 0$$ 3. **Solution:** $$x = 5$$ 4. **Axis of symmetry:** $$x = 5$$ 5. **Vertex:** $$y = 0$$ Vertex = (5, 0)$$ 6. **Y-intercept:** $$y = 25$$ 7. **X-intercepts:** $$x = 5$$ --- 1. **Problem:** Analyze $4x^2 + 5x + 20$ (no equation given, so analyze the function). 2. **Axis of symmetry:** $$x = -\frac{5}{2 \times 4} = -\frac{5}{8} = -0.625$$ 3. **Vertex:** $$y = 4(-0.625)^2 + 5(-0.625) + 20 = 4(0.390625) - 3.125 + 20 = 1.5625 - 3.125 + 20 = 18.4375$$ Vertex = (-0.625, 18.4375)$$ 4. **Y-intercept:** $$y = 20$$ 5. **X-intercepts:** Use quadratic formula: $$a=4, b=5, c=20$$ $$\Delta = 5^2 - 4(4)(20) = 25 - 320 = -295 < 0$$ No real x-intercepts. --- **Summary:** - Each quadratic's axis of symmetry, vertex, y-intercept, and x-intercepts are found. - Solutions for x-intercepts are shown using factoring or quadratic formula. - Graphs are parabolas opening up or down depending on $a$.