1. **Identify the values of a, b, and c for each quadratic equation:** - a. For $3x^2 + 8x + 4 = 0$, $a=3$, $b=8$, $c=4$. - b. For $2x^2 - 5x + 2 = 0$, $a=2$, $b=-5$, $c=2$. - c. For $-9x^2 + 13x - 1 = 0$, $a=-9$, $b=13$, $c=-1$. - d. For $x^2 + x - 11 = 0$, $a=1$, $b=1$, $c=-11$. - e. For $-x^2 + 16x + 64 = 0$, $a=-1$, $b=16$, $c=64$. 2. **Use the quadratic formula to verify the solutions:** The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ **a. Verify solutions for $x^2 + 9x + 20 = 0$ where $x = -4$ and $x = -5$:** - Here, $a=1$, $b=9$, $c=20$. - Calculate the discriminant: $$b^2 - 4ac = 9^2 - 4(1)(20) = 81 - 80 = 1$$ - Substitute into the formula: $$x = \frac{-9 \pm \sqrt{1}}{2(1)} = \frac{-9 \pm 1}{2}$$ - Calculate each root: $$x_1 = \frac{-9 + 1}{2} = \frac{-8}{2} = -4$$ $$x_2 = \frac{-9 - 1}{2} = \frac{-10}{2} = -5$$ These match the given solutions. **b. Verify solutions for $x^2 - 10x + 21 = 0$ where $x = 3$ and $x = 7$:** - Here, $a=1$, $b=-10$, $c=21$. - Calculate the discriminant: $$b^2 - 4ac = (-10)^2 - 4(1)(21) = 100 - 84 = 16$$ - Substitute into the formula: $$x = \frac{-(-10) \pm \sqrt{16}}{2(1)} = \frac{10 \pm 4}{2}$$ - Calculate each root: $$x_1 = \frac{10 + 4}{2} = \frac{14}{2} = 7$$ $$x_2 = \frac{10 - 4}{2} = \frac{6}{2} = 3$$ These match the given solutions. Thus, the quadratic formula confirms the solutions are correct.