Question

Rezolvati ecuatiile urmatoare cu delta!
x la a doua +9x+20=0
x la a doua +11x+30=0
x la a doua +4x+49=0
x la a doua +x +1=0
Va multumesc!

Answer 1

$\star) {x}^{2} + 9x + 20 = 0$

$a = 1$

$b = 9$

$c = 20$

$\Delta = {b}^{2} - 4ac$

$\Delta = {9}^{2} - 4 \times 1 \times 20$

$\Delta = 81 - 80$

$\Delta = 1$

$\Delta > 0 => \exists \:\: x_{1} \neq x_{2} \: \in \:\mathbb{R}$

$x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}$

$x_{1,2}=\frac{-9\pm\sqrt{1}}{2 \times 1}$

$x_{1,2}=\frac{-9\pm1}{2}$

$x_{1}=\frac{-9 + 1}{2}$

$x_{1}= - \frac{8}{2}$

$x_{1}= - 4$

$x_{2}=\frac{-9 - 1}{2}$

$x_{2}= - \frac{10}{2}$

$x_{2}= - 5$

$\star) {x}^{2} + 11x + 30 = 0$

$a = 1$

$b = 11$

$c = 30$

$\Delta = {b}^{2} - 4ac$

$\Delta = {11}^{2} - 4 \times 1 \times 30$

$\Delta = 121 - 120$

$\Delta = 1$

$\Delta > 0 => \exists \: \: x_{1} \neq x_{2} \: \in \:\mathbb{R}$

$x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}$

$x_{1,2}=\frac{-11\pm\sqrt{1}}{2 \times 1}$

$x_{1,2}=\frac{-11\pm1}{2}$

$x_{1}=\frac{-11 + 1}{2}$

$x_{1}= - \frac{10}{2}$

$x_{1}= - 5$

$x_{2}= \frac{ - 11 - 1}{2}$

$x_{2}= - \frac{12}{2}$

$x_{2}= - 6$

$\star) {x}^{2} + 4x + 49 = 0$

$a = 1$

$b = 4$

$c = 49$

$\Delta = {b}^{2} - 4ac$

$\Delta = {4}^{2} - 4 \times 1 \times 49$

$\Delta = 16 - 196$

$\Delta = - 180$

$\Delta < 0 = > \nexists \: \: x_{1},x_{2}\:\in\:\mathbb{R}$

$= > \exists \: \: x_{1},x_{2}\:\in\:\mathbb{C}$

$x_{1,2}=\frac{-b\pm i\sqrt{ - \Delta}}{2a}$

$x_{1,2}=\frac{-4\pm i\sqrt{ - ( - 180)}}{2 \times 1}$

$x_{1,2}=\frac{-4\pm i\sqrt{ 180}}{2}$

$x_{1}=\frac{-4 + i\sqrt{ 180}}{2}$

$x_{1}=\frac{-4 + 6\sqrt{5}i }{2}$

$x_{1}=\frac{2( - 2 + 3\sqrt{5}i )}{2}$

$x_{1}= - 2 + 3\sqrt{5}i$

$x_{2}=- 2 - 3\sqrt{5}i$

$\star) {x}^{2} + x + 1 = 0$

$a = 1$

$b = 1$

$c = 1$

$\Delta = {b}^{2} - 4ac$

$\Delta = {1}^{2} - 4 \times 1 \times 1$

$\Delta = 1 - 4$

$\Delta = - 3$

$\Delta < 0 = > \nexists \: \: x_{1},x_{2}\:\in\:\mathbb{R}$

$= > \exists \: \: x_{1},x_{2}\:\in\:\mathbb{C}$

$x_{1,2}=\frac{-b\pm i\sqrt{ - \Delta}}{2a}$

$x_{1,2}=\frac{-1\pm i\sqrt{ - ( - 3)}}{2 \times 1}$

$x_{1,2}=\frac{-1\pm i\sqrt{ 3}}{2 }$

$x_{1}=\frac{-1 + i\sqrt{ 3}}{2 }$

$x_{1}=-\frac{1 - i\sqrt{ 3}}{2 }$

$x_{2}=\frac{-1 - i\sqrt{ 3}}{2 }$

$x_{2}=-\frac{1 +i\sqrt{ 3}}{2 }$