1. **Problem:** Solve and analyze the quadratic equation $x^2 - 25 = 0$. 2. **Formula:** Quadratic equation standard form is $ax^2 + bx + c = 0$. 3. **Step 1:** Solve for $x$: $$x^2 - 25 = 0 \implies x^2 = 25 \implies x = \pm 5$$ 4. **Axis of symmetry:** $x = -\frac{b}{2a} = -\frac{0}{2 \times 1} = 0$ 5. **Vertex:** Substitute $x=0$ into the equation: $$y = 0^2 - 25 = -25$$ Vertex is $(0, -25)$. 6. **Y-intercept:** When $x=0$, $y = -25$. 7. **X-intercepts:** $(5,0)$ and $(-5,0)$. --- 1. **Problem:** Solve and analyze $-x^2 - 4x = 0$. 2. **Step 1:** Factor: $$-x^2 - 4x = -x(x + 4) = 0$$ 3. **Solutions:** $x=0$ or $x=-4$. 4. **Axis of symmetry:** $x = -\frac{b}{2a} = -\frac{-4}{2 \times (-1)} = -\frac{-4}{-2} = -2$ 5. **Vertex:** Substitute $x=-2$: $$y = -(-2)^2 - 4(-2) = -4 + 8 = 4$$ Vertex is $(-2, 4)$. 6. **Y-intercept:** $x=0$, $y=0$. 7. **X-intercepts:** $(0,0)$ and $(-4,0)$. --- 1. **Problem:** Solve and analyze $-2x^2 + 5x + 6 = 0$. 2. **Step 1:** 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. **Solutions:** $$x_1 = \frac{-5 + \sqrt{73}}{-4}, \quad x_2 = \frac{-5 - \sqrt{73}}{-4}$$ 4. **Axis of symmetry:** $$x = -\frac{b}{2a} = -\frac{5}{2 \times (-2)} = \frac{5}{4} = 1.25$$ 5. **Vertex:** Substitute $x=1.25$: $$y = -2(1.25)^2 + 5(1.25) + 6 = -2(1.5625) + 6.25 + 6 = -3.125 + 12.25 = 9.125$$ Vertex is $(1.25, 9.125)$. 6. **Y-intercept:** $x=0$, $y=6$. 7. **X-intercepts:** Approximate: $$x_1 \approx \frac{-5 + 8.544}{-4} = \frac{3.544}{-4} = -0.886$$ $$x_2 \approx \frac{-5 - 8.544}{-4} = \frac{-13.544}{-4} = 3.386$$ --- 1. **Problem:** Solve and analyze $x^2 - 10x + 25 = 0$. 2. **Step 1:** Recognize perfect square: $$x^2 - 10x + 25 = (x - 5)^2 = 0$$ 3. **Solution:** $x=5$ (double root). 4. **Axis of symmetry:** $x = 5$. 5. **Vertex:** Substitute $x=5$: $$y = 0$$ Vertex is $(5,0)$. 6. **Y-intercept:** $x=0$, $y=25$. 7. **X-intercept:** $(5,0)$. --- 1. **Problem:** Analyze $4x^2 + 5x + 20$ (no equation given, so analyze graph). 2. **Axis of symmetry:** $$x = -\frac{b}{2a} = -\frac{5}{2 \times 4} = -\frac{5}{8} = -0.625$$ 3. **Vertex:** Substitute $x=-0.625$: $$y = 4(-0.625)^2 + 5(-0.625) + 20 = 4(0.390625) - 3.125 + 20 = 1.5625 - 3.125 + 20 = 18.4375$$ Vertex is $(-0.625, 18.4375)$. 4. **Y-intercept:** $x=0$, $y=20$. 5. **X-intercepts:** Calculate discriminant: $$\Delta = b^2 - 4ac = 25 - 4(4)(20) = 25 - 320 = -295 < 0$$ No real roots, so no x-intercepts. --- **Summary:** - Each quadratic was solved for roots. - Axis of symmetry and vertex found using formulas. - Y-intercepts found by substituting $x=0$. - X-intercepts found by solving the equation or noting no real roots.