1. Solve $x^2 - 25 = 0$. - This is a difference of squares: $a^2 - b^2 = (a-b)(a+b)$. - Factor: $(x-5)(x+5) = 0$. - Set each factor to zero: $x-5=0 \Rightarrow x=5$, $x+5=0 \Rightarrow x=-5$. - Roots: $x=5$ and $x=-5$. 2. Solve $-x^2 - 4x = 0$. - Factor out $-x$: $-x(x+4) = 0$. - Set each factor to zero: $-x=0 \Rightarrow x=0$, $x+4=0 \Rightarrow x=-4$. - Roots: $x=0$ and $x=-4$. 3. Solve $-2x^2 + 5x + 6 = 0$. - Use quadratic formula: $x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=-2$, $b=5$, $c=6$. - Calculate discriminant: $\Delta = 5^2 - 4(-2)(6) = 25 + 48 = 73$. - Roots: $x=\frac{-5 \pm \sqrt{73}}{2(-2)} = \frac{-5 \pm \sqrt{73}}{-4}$. - Simplify signs: $x=\frac{5 \mp \sqrt{73}}{4}$. 4. Solve $x^2 - 10x + 25 = 0$. - Recognize perfect square: $(x-5)^2 = 0$. - Root: $x=5$ (double root). 5. Solve $4x^2 + 5x + 20 = 0$. - Calculate discriminant: $\Delta = 5^2 - 4(4)(20) = 25 - 320 = -295$. - Since $\Delta < 0$, no real roots; graph does not cross x-axis. Summary: - 1: Roots at $x=\pm5$. - 2: Roots at $x=0, -4$. - 3: Roots at $x=\frac{5 \mp \sqrt{73}}{4}$. - 4: Root at $x=5$ (double root). - 5: No real roots. Each quadratic graph is a parabola; the sign of $a$ determines if it opens up ($a>0$) or down ($a<0$).