Răspuns :
[tex] \star) {x}^{2} + 9x + 20 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 9[/tex]
[tex]c = 20[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {9}^{2} - 4 \times 1 \times 20[/tex]
[tex]\Delta = 81 - 80[/tex]
[tex]\Delta = 1[/tex]
[tex]\Delta > 0 => \exists \:\: x_{1} \neq x_{2} \: \in \:\mathbb{R}[/tex]
[tex]x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-9\pm\sqrt{1}}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{-9\pm1}{2}[/tex]
[tex]x_{1}=\frac{-9 + 1}{2}[/tex]
[tex]x_{1}= - \frac{8}{2}[/tex]
[tex]x_{1}= - 4[/tex]
[tex]x_{2}=\frac{-9 - 1}{2}[/tex]
[tex]x_{2}= - \frac{10}{2}[/tex]
[tex]x_{2}= - 5[/tex]
[tex] \star) {x}^{2} + 11x + 30 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 11[/tex]
[tex]c = 30[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {11}^{2} - 4 \times 1 \times 30[/tex]
[tex]\Delta = 121 - 120[/tex]
[tex]\Delta = 1[/tex]
[tex]\Delta > 0 => \exists \: \: x_{1} \neq x_{2} \: \in \:\mathbb{R}[/tex]
[tex]x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-11\pm\sqrt{1}}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{-11\pm1}{2}[/tex]
[tex]x_{1}=\frac{-11 + 1}{2}[/tex]
[tex]x_{1}= - \frac{10}{2}[/tex]
[tex]x_{1}= - 5[/tex]
[tex]x_{2}= \frac{ - 11 - 1}{2}[/tex]
[tex]x_{2}= - \frac{12}{2}[/tex]
[tex]x_{2}= - 6[/tex]
[tex] \star) {x}^{2} + 4x + 49 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 4[/tex]
[tex]c = 49[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {4}^{2} - 4 \times 1 \times 49[/tex]
[tex]\Delta = 16 - 196[/tex]
[tex]\Delta = - 180 [/tex]
[tex]\Delta < 0 = > \nexists \: \: x_{1},x_{2}\:\in\:\mathbb{R}[/tex]
[tex] = > \exists \: \: x_{1},x_{2}\:\in\:\mathbb{C}[/tex]
[tex]x_{1,2}=\frac{-b\pm i\sqrt{ - \Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-4\pm i\sqrt{ - ( - 180)}}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{-4\pm i\sqrt{ 180}}{2}[/tex]
[tex]x_{1}=\frac{-4 + i\sqrt{ 180}}{2}[/tex]
[tex]x_{1}=\frac{-4 + 6\sqrt{5}i }{2}[/tex]
[tex]x_{1}=\frac{2( - 2 + 3\sqrt{5}i )}{2}[/tex]
[tex]x_{1}= - 2 + 3\sqrt{5}i [/tex]
[tex]x_{2}=- 2 - 3\sqrt{5}i [/tex]
[tex] \star) {x}^{2} + x + 1 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 1[/tex]
[tex]c = 1[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {1}^{2} - 4 \times 1 \times 1[/tex]
[tex]\Delta = 1 - 4[/tex]
[tex]\Delta = - 3[/tex]
[tex]\Delta < 0 = > \nexists \: \: x_{1},x_{2}\:\in\:\mathbb{R}[/tex]
[tex] = > \exists \: \: x_{1},x_{2}\:\in\:\mathbb{C}[/tex]
[tex]x_{1,2}=\frac{-b\pm i\sqrt{ - \Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-1\pm i\sqrt{ - ( - 3)}}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{-1\pm i\sqrt{ 3}}{2 }[/tex]
[tex]x_{1}=\frac{-1 + i\sqrt{ 3}}{2 }[/tex]
[tex]x_{1}=-\frac{1 - i\sqrt{ 3}}{2 }[/tex]
[tex]x_{2}=\frac{-1 - i\sqrt{ 3}}{2 }[/tex]
[tex]x_{2}=-\frac{1 +i\sqrt{ 3}}{2 }[/tex]
[tex]a = 1[/tex]
[tex]b = 9[/tex]
[tex]c = 20[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {9}^{2} - 4 \times 1 \times 20[/tex]
[tex]\Delta = 81 - 80[/tex]
[tex]\Delta = 1[/tex]
[tex]\Delta > 0 => \exists \:\: x_{1} \neq x_{2} \: \in \:\mathbb{R}[/tex]
[tex]x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-9\pm\sqrt{1}}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{-9\pm1}{2}[/tex]
[tex]x_{1}=\frac{-9 + 1}{2}[/tex]
[tex]x_{1}= - \frac{8}{2}[/tex]
[tex]x_{1}= - 4[/tex]
[tex]x_{2}=\frac{-9 - 1}{2}[/tex]
[tex]x_{2}= - \frac{10}{2}[/tex]
[tex]x_{2}= - 5[/tex]
[tex] \star) {x}^{2} + 11x + 30 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 11[/tex]
[tex]c = 30[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {11}^{2} - 4 \times 1 \times 30[/tex]
[tex]\Delta = 121 - 120[/tex]
[tex]\Delta = 1[/tex]
[tex]\Delta > 0 => \exists \: \: x_{1} \neq x_{2} \: \in \:\mathbb{R}[/tex]
[tex]x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-11\pm\sqrt{1}}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{-11\pm1}{2}[/tex]
[tex]x_{1}=\frac{-11 + 1}{2}[/tex]
[tex]x_{1}= - \frac{10}{2}[/tex]
[tex]x_{1}= - 5[/tex]
[tex]x_{2}= \frac{ - 11 - 1}{2}[/tex]
[tex]x_{2}= - \frac{12}{2}[/tex]
[tex]x_{2}= - 6[/tex]
[tex] \star) {x}^{2} + 4x + 49 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 4[/tex]
[tex]c = 49[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {4}^{2} - 4 \times 1 \times 49[/tex]
[tex]\Delta = 16 - 196[/tex]
[tex]\Delta = - 180 [/tex]
[tex]\Delta < 0 = > \nexists \: \: x_{1},x_{2}\:\in\:\mathbb{R}[/tex]
[tex] = > \exists \: \: x_{1},x_{2}\:\in\:\mathbb{C}[/tex]
[tex]x_{1,2}=\frac{-b\pm i\sqrt{ - \Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-4\pm i\sqrt{ - ( - 180)}}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{-4\pm i\sqrt{ 180}}{2}[/tex]
[tex]x_{1}=\frac{-4 + i\sqrt{ 180}}{2}[/tex]
[tex]x_{1}=\frac{-4 + 6\sqrt{5}i }{2}[/tex]
[tex]x_{1}=\frac{2( - 2 + 3\sqrt{5}i )}{2}[/tex]
[tex]x_{1}= - 2 + 3\sqrt{5}i [/tex]
[tex]x_{2}=- 2 - 3\sqrt{5}i [/tex]
[tex] \star) {x}^{2} + x + 1 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 1[/tex]
[tex]c = 1[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {1}^{2} - 4 \times 1 \times 1[/tex]
[tex]\Delta = 1 - 4[/tex]
[tex]\Delta = - 3[/tex]
[tex]\Delta < 0 = > \nexists \: \: x_{1},x_{2}\:\in\:\mathbb{R}[/tex]
[tex] = > \exists \: \: x_{1},x_{2}\:\in\:\mathbb{C}[/tex]
[tex]x_{1,2}=\frac{-b\pm i\sqrt{ - \Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-1\pm i\sqrt{ - ( - 3)}}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{-1\pm i\sqrt{ 3}}{2 }[/tex]
[tex]x_{1}=\frac{-1 + i\sqrt{ 3}}{2 }[/tex]
[tex]x_{1}=-\frac{1 - i\sqrt{ 3}}{2 }[/tex]
[tex]x_{2}=\frac{-1 - i\sqrt{ 3}}{2 }[/tex]
[tex]x_{2}=-\frac{1 +i\sqrt{ 3}}{2 }[/tex]
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