\( \begin{array}{ll}\text { (9) } 4 x^{2}-4 x-3=0 & \text { (10) } 3 x^{2}+8 x=-2 \text { (two decima } \\ \text { (11) }-5 x^{2}-2 x+1=0 \text { (two decimals) } & \text { (12) }-3 x^{2}-11 x=9 \text { (two decim } \\ \text { (13) } 3 x^{2}+33 x+30=0 & \text { (14) } 7-12 x-x^{2}=0 \text { (surd forn } \\ \text { (15) } 16 x+12-x^{2}=0 \text { (two decimals) } & \text { (16) } \frac{2}{3} x^{2}-\frac{5}{2} x+\frac{1}{4}=0 \text { (surd fo }\end{array} \)

Sure! Let's solve each of the quadratic equations from (9) to (16), providing solutions either rounded to two decimal places or in surd (exact radical) form as specified. --- ### **(9) \(4x^{2} - 4x - 3 = 0\)** **Solution Method:** Quadratic Formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = 4\), \(b = -4\), and \(c = -3\). \[ x = \frac{4 \pm \sqrt{16 + 48}}{8} = \frac{4 \pm \sqrt{64}}{8} = \frac{4 \pm 8}{8} \] \[ \Rightarrow x = \frac{12}{8} = 1.5 \quad \text{and} \quad x = \frac{-4}{8} = -0.5 \] **Solutions:** \(x = 1.5\) and \(x = -0.5\) --- ### **(10) \(3x^{2} + 8x = -2\)** **Rearranged Equation:** \(3x^{2} + 8x + 2 = 0\) **Solution Method:** Quadratic Formula (Rounded to two decimal places) \[ x = \frac{-8 \pm \sqrt{64 - 24}}{6} = \frac{-8 \pm \sqrt{40}}{6} = \frac{-8 \pm 6.3246}{6} \] \[ x_1 = \frac{-8 + 6.3246}{6} \approx -0.28 \quad \text{and} \quad x_2 = \frac{-8 - 6.3246}{6} \approx -2.39 \] **Solutions:** \(x \approx -0.28\) and \(x \approx -2.39\) --- ### **(11) \(-5x^{2} - 2x + 1 = 0\)** **Rearranged Equation:** \(5x^{2} + 2x - 1 = 0\) **Solution Method:** Quadratic Formula (Rounded to two decimal places) \[ x = \frac{-2 \pm \sqrt{4 + 20}}{10} = \frac{-2 \pm \sqrt{24}}{10} = \frac{-2 \pm 4.899}{10} \] \[ x_1 = \frac{-2 + 4.899}{10} \approx 0.29 \quad \text{and} \quad x_2 = \frac{-2 - 4.899}{10} \approx -0.69 \] **Solutions:** \(x \approx 0.29\) and \(x \approx -0.69\) --- ### **(12) \(-3x^{2} - 11x = 9\)** **Rearranged Equation:** \(-3x^{2} - 11x - 9 = 0\) **Simplified Equation:** \(3x^{2} + 11x + 9 = 0\) **Solution Method:** Quadratic Formula (Rounded to two decimal places) \[ x = \frac{-11 \pm \sqrt{121 - 108}}{6} = \frac{-11 \pm \sqrt{13}}{6} \] \[ \sqrt{13} \approx 3.6056 \] \[ x_1 = \frac{-11 + 3.6056}{6} \approx -1.23 \quad \text{and} \quad x_2 = \frac{-11 - 3.6056}{6} \approx -2.43 \] **Solutions:** \(x \approx -1.23\) and \(x \approx -2.43\) --- ### **(13) \(3x^{2} + 33x + 30 = 0\)** **Solution Method:** Quadratic Formula \[ x = \frac{-33 \pm \sqrt{1089 - 360}}{6} = \frac{-33 \pm \sqrt{729}}{6} = \frac{-33 \pm 27}{6} \] \[ x_1 = \frac{-33 + 27}{6} = \frac{-6}{6} = -1 \quad \text{and} \quad x_2 = \frac{-33 - 27}{6} = \frac{-60}{6} = -10 \] **Solutions:** \(x = -1\) and \(x = -10\) --- ### **(14) \(7 - 12x - x^{2} = 0\)** **Rearranged Equation:** \(-x^{2} - 12x + 7 = 0\) **Simplified Equation:** \(x^{2} + 12x - 7 = 0\) **Solution Method:** Quadratic Formula (Expressed in Surd Form) \[ x = \frac{-12 \pm \sqrt{144 + 28}}{2} = \frac{-12 \pm \sqrt{172}}{2} = \frac{-12 \pm 2\sqrt{43}}{2} \] \[ x = -6 \pm \sqrt{43} \] **Solutions:** \(x = -6 + \sqrt{43}\) and \(x = -6 - \sqrt{43}\) --- ### **(15) \(16x + 12 - x^{2} = 0\)** **Rearranged Equation:** \(-x^{2} + 16x + 12 = 0\) **Simplified Equation:** \(x^{2} - 16x - 12 = 0\) **Solution Method:** Quadratic Formula (Rounded to two decimal places) \[ x = \frac{16 \pm \sqrt{256 + 48}}{2} = \frac{16 \pm \sqrt{304}}{2} \approx \frac{16 \pm 17.4356}{2} \] \[ x_1 = \frac{16 + 17.4356}{2} \approx 16.72 \quad \text{and} \quad x_2 = \frac{16 - 17.4356}{2} \approx -0.72 \] **Solutions:** \(x \approx 16.72\) and \(x \approx -0.72\) --- ### **(16) \(\frac{2}{3}x^{2} - \frac{5}{2}x + \frac{1}{4} = 0\)** **Solution Method:** Quadratic Formula (Expressed in Surd Form) First, eliminate fractions by multiplying the entire equation by 12: \[ 12 \cdot \left(\frac{2}{3}x^{2} - \frac{5}{2}x + \frac{1}{4}\right) = 0 \Rightarrow 8x^{2} - 30x + 3 = 0 \] Now, apply the quadratic formula: \[ x = \frac{30 \pm \sqrt{900 - 96}}{16} = \frac{30 \pm \sqrt{804}}{16} = \frac{30 \pm 2\sqrt{201}}{16} = \frac{15 \pm \sqrt{201}}{8} \] **Solutions:** \(x = \dfrac{15 + \sqrt{201}}{8}\) and \(x = \dfrac{15 - \sqrt{201}}{8}\) --- Feel free to ask if you need further explanations or assistance with any of these solutions!