1. **Problem statement:** Solve the quadratic equations using the quadratic formula and round answers to 1 decimal place. 2. **Quadratic formula:** For equation $ax^2 + bx + c = 0$, solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Important: The discriminant $\Delta = b^2 - 4ac$ determines the nature of roots. 3. **Solve each equation:** (a) $x^2 + 5x + 1 = 0$ with $a=1$, $b=5$, $c=1$ Calculate discriminant: $$\Delta = 5^2 - 4 \times 1 \times 1 = 25 - 4 = 21$$ Apply formula: $$x = \frac{-5 \pm \sqrt{21}}{2 \times 1} = \frac{-5 \pm \sqrt{21}}{2}$$ Approximate: $$x_1 = \frac{-5 + 4.5826}{2} = \frac{-0.4174}{2} = -0.2$$ $$x_2 = \frac{-5 - 4.5826}{2} = \frac{-9.5826}{2} = -4.8$$ (b) $2x^2 + 7x + 2 = 0$ with $a=2$, $b=7$, $c=2$ $$\Delta = 7^2 - 4 \times 2 \times 2 = 49 - 16 = 33$$ $$x = \frac{-7 \pm \sqrt{33}}{2 \times 2} = \frac{-7 \pm 5.7446}{4}$$ $$x_1 = \frac{-7 + 5.7446}{4} = \frac{-1.2554}{4} = -0.3$$ $$x_2 = \frac{-7 - 5.7446}{4} = \frac{-12.7446}{4} = -3.2$$ (c) $4x^2 + 8x + 3 = 0$ with $a=4$, $b=8$, $c=3$ $$\Delta = 8^2 - 4 \times 4 \times 3 = 64 - 48 = 16$$ $$x = \frac{-8 \pm \sqrt{16}}{2 \times 4} = \frac{-8 \pm 4}{8}$$ $$x_1 = \frac{-8 + 4}{8} = \frac{-4}{8} = -0.5$$ $$x_2 = \frac{-8 - 4}{8} = \frac{-12}{8} = -1.5$$ (d) $x^2 + 2x - 4 = 0$ with $a=1$, $b=2$, $c=-4$ $$\Delta = 2^2 - 4 \times 1 \times (-4) = 4 + 16 = 20$$ $$x = \frac{-2 \pm \sqrt{20}}{2} = \frac{-2 \pm 4.4721}{2}$$ $$x_1 = \frac{-2 + 4.4721}{2} = \frac{2.4721}{2} = 1.2$$ $$x_2 = \frac{-2 - 4.4721}{2} = \frac{-6.4721}{2} = -3.2$$ (e) $3x^2 + 4x - 5 = 0$ with $a=3$, $b=4$, $c=-5$ $$\Delta = 4^2 - 4 \times 3 \times (-5) = 16 + 60 = 76$$ $$x = \frac{-4 \pm \sqrt{76}}{2 \times 3} = \frac{-4 \pm 8.7178}{6}$$ $$x_1 = \frac{-4 + 8.7178}{6} = \frac{4.7178}{6} = 0.8$$ $$x_2 = \frac{-4 - 8.7178}{6} = \frac{-12.7178}{6} = -2.1$$ (f) $2x^2 + 5x - 10 = 0$ with $a=2$, $b=5$, $c=-10$ $$\Delta = 5^2 - 4 \times 2 \times (-10) = 25 + 80 = 105$$ $$x = \frac{-5 \pm \sqrt{105}}{4} = \frac{-5 \pm 10.2469}{4}$$ $$x_1 = \frac{-5 + 10.2469}{4} = \frac{5.2469}{4} = 1.3$$ $$x_2 = \frac{-5 - 10.2469}{4} = \frac{-15.2469}{4} = -3.8$$ **Final answers:** (a) $x = -0.2, -4.8$ (b) $x = -0.3, -3.2$ (c) $x = -0.5, -1.5$ (d) $x = 1.2, -3.2$ (e) $x = 0.8, -2.1$ (f) $x = 1.3, -3.8$