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