[tex]a) {x}^{2} - 2x = 8[/tex]
[tex] {x}^{2} - 2x - 8 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = - 2[/tex]
[tex]c = - 8[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {( - 2)}^{2} - 4 \times 1 \times ( - 8)[/tex]
[tex]\Delta = 4 + 32[/tex]
[tex]\Delta = 36 > 0 = > \exists \: \: x_{1} \neq \: x_{2}[/tex]
[tex]x_{1,2}=\frac{-b \: \pm \: \sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-( - 2)\pm \sqrt{36} }{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{2\pm6}{2}[/tex]
[tex]x_{1}=\frac{2 + 6}{2} = \frac{8}{2} = 4[/tex]
[tex]x_{2}=\frac{2 - 6}{2} = - \frac{4}{2} = - 2[/tex]
[tex]b) {x}^{2} + 4x = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 4[/tex]
[tex]c = 0[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {4}^{2} - 4 \times 1 \times 0[/tex]
[tex]\Delta = 16 > 0 = > \exists \: \: x_{1} \neq \: x_{2}[/tex]
[tex]x_{1,2} = \frac{ - b \pm \sqrt{\Delta} }{2a} [/tex]
[tex]x_{1,2} = \frac{ - 4 \pm \sqrt{16} }{2 \times 1} [/tex]
[tex]x_{1,2} = \frac{ - 4 \pm4}{2} [/tex]
[tex]x_{1} = \frac{ - 4 + 4}{2} = \frac{0}{2} = 0[/tex]
[tex]x_{2} = \frac{ - 4 - 4}{2} = \frac{ - 8}{2} = - 4[/tex]
[tex]c) {x}^{2} + 5 = 6x[/tex]
[tex] {x}^{2} - 6x + 5 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = - 6[/tex]
[tex]c = 5[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {( - 6)}^{2} - 4 \times 1 \times 5 = 36 - 20 = 16 > 0 = > \exists \: x_{1} \neq \: x_{2}[/tex]
[tex]x_{1,2}=\frac{-b \: \pm \: \sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-( - 6)\pm\sqrt{16}}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{6\pm4}{2}[/tex]
[tex]x_{1}= \frac{6 + 4}{2} = \frac{10}{2} = 5[/tex]
[tex]x_{2}= \frac{6 - 4}{2} = \frac{2}{2} = 1[/tex]
[tex]d)9 {x}^{2} + 6x = 24[/tex]
[tex]9 {x}^{2} + 6x - 24 = 0[/tex]
[tex]a = 9[/tex]
[tex]b = 6[/tex]
[tex]c = - 24[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {6}^{2} - 4 \times 9 \times ( - 24)[/tex]
[tex]\Delta = 36 + 864 = 900 > 0 = > \exists \: x_{1} \: \neq \: x_{2}[/tex]
[tex]x_{1,2}=\frac{-b \: \pm \: \sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-6\pm\sqrt{900}}{2 \times 9}[/tex]
[tex]x_{1,2}=\frac{-6\pm30}{18}[/tex]
[tex]x_{1}=\frac{-6 + 30}{18} = \frac{24}{18} = \frac{4}{3} [/tex]
[tex]x_{2}=\frac{-6 - 30}{18} = - \frac{36}{18} = - 2[/tex]
[tex]e)4 {x}^{2} - 48 = 4x[/tex]
[tex]4 {x}^{2} - 4x - 48 = 0[/tex]
[tex]a = 4[/tex]
[tex]b = - 4[/tex]
[tex]c = - 48[/tex]
[tex]\Delta = {( - 4)}^{2} - 4 \times 4 \times ( - 48)[/tex]
[tex]\Delta = 16 + 768[/tex]
[tex]\Delta = 784 > 0 = > \exists \: x_{1} \: \neq \: x_{2}[/tex]
[tex]x_{1,2}=\frac{-b \: \pm \: \sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-(-4) \: \pm\sqrt{784}}{2\times4}[/tex]
[tex]x_{1,2}=\frac{4\pm28}{8}[/tex]
[tex]x_{1}=\frac{4 + 28}{8} = \frac{32}{8} = 4[/tex]
[tex]x_{2}=\frac{4 - 28}{8} = - \frac{24}{8} = - 3[/tex]
[tex]f)16 {x}^{2} + 8x = 80[/tex]
[tex]16 {x}^{2} + 8x - 80 = 0[/tex]
[tex]a = 16[/tex]
[tex]b = 8[/tex]
[tex]c = - 80[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {8}^{2} - 4 \times 16 \times ( - 80)[/tex]
[tex]\Delta = 64 + 5120[/tex]
[tex]\Delta = 5184 > 0 = > \exists \: x_{1} \: \neq \: x_{2}[/tex]
[tex]x_{1,2}=\frac{-b \: \pm \: \sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-8\pm\sqrt{5184}}{2 \times 16}[/tex]
[tex]x_{1,2}=\frac{-8\pm72}{32}[/tex]
[tex]x_{1} = \frac{ - 8 + 72}{32} = \frac{64}{32} = 2[/tex]
[tex]x_{2} = \frac{ - 8 - 72}{32} = - \frac{80}{32} = - \frac{10}{4} = - \frac{5}{2} [/tex]
