Question

Rezolvati in R urmaătoarele ecuatii:

Answer 1

a) x² - 2x = 8
x² - 2x - 8 = 0
x² + 2x - 4x - 8 = 0
x( x + 2 ) - 4( x + 2 ) =0
( x + 2)( x - 4 ) = 0  
x + 2 = 0 ⇒ x = -2
x - 4 = 0 ⇒ x = 4 

b) x² + 4x = 0
x( x + 4 ) = 0 ⇒ x = 0
x + 4 = 0 ⇒ x = -4 

c) x² + 5 = 6x 
x² + 5 - 6x = 0
x² - 6x + 5 = 0
x² - x - 5x + 5 = 0
x( x - 1) - 5( x - 1) =0
( x - 1 )( x - 5 ) = 0
x - 1 = 0 ⇒ x = 1
x - 5 = 0 ⇒ x = 5 

d) 9x² + 6x = 24
9x² + 6x - 24 = 0   / : 3
3x² + 2x - 8 = 0
3x² + 6x - 4x - 8 = 0
3x( x + 2 )- 4( x + 2 ) = 0
( x + 2 )( 3x - 4 ) = 0 
x + 2 = 0 ⇒ x = -2
3x - 4 = 0 ⇒ x = 4/3 


e) 4x² - 48 = 4x
4x² - 48 - 4x = 0  / : 4
x² - x - 12 = 0 
x² + 3x - 4x - 12 = 0
x( x + 3 )( x - 4 ) = 0
( x + 3 )( x - 4 ) = 0 
x + 3 = 0 ⇒ x = -3
x - 4 = 0 ⇒ x = 4 

d) 16x² + 8x = 80
16x² + 8x - 80 = 0 / : 8
2x² + x - 10 = 0
2x² + 5x - 4x - 10 = 0
x( 2x + 5 ) - 2( 2x + 5 ) = 0
( 2x +5 )( x - 2 ) =0 
2x + 5 = 0 ⇒  x = - 5/2
x - 2 = 0 ⇒ x = 2 

Answer 2

$a) {x}^{2} - 2x = 8$

${x}^{2} - 2x - 8 = 0$

$a = 1$

$b = - 2$

$c = - 8$

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

$\Delta = {( - 2)}^{2} - 4 \times 1 \times ( - 8)$

$\Delta = 4 + 32$

$\Delta = 36 > 0 = > \exists \: \: x_{1} \neq \: x_{2}$

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

$x_{1,2}=\frac{-( - 2)\pm \sqrt{36} }{2 \times 1}$

$x_{1,2}=\frac{2\pm6}{2}$

$x_{1}=\frac{2 + 6}{2} = \frac{8}{2} = 4$

$x_{2}=\frac{2 - 6}{2} = - \frac{4}{2} = - 2$

$b) {x}^{2} + 4x = 0$

$a = 1$

$b = 4$

$c = 0$

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

$\Delta = {4}^{2} - 4 \times 1 \times 0$

$\Delta = 16 > 0 = > \exists \: \: x_{1} \neq \: x_{2}$

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

$x_{1,2} = \frac{ - 4 \pm \sqrt{16} }{2 \times 1}$

$x_{1,2} = \frac{ - 4 \pm4}{2}$

$x_{1} = \frac{ - 4 + 4}{2} = \frac{0}{2} = 0$

$x_{2} = \frac{ - 4 - 4}{2} = \frac{ - 8}{2} = - 4$

$c) {x}^{2} + 5 = 6x$

${x}^{2} - 6x + 5 = 0$

$a = 1$

$b = - 6$

$c = 5$

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

$\Delta = {( - 6)}^{2} - 4 \times 1 \times 5 = 36 - 20 = 16 > 0 = > \exists \: x_{1} \neq \: x_{2}$

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

$x_{1,2}=\frac{-( - 6)\pm\sqrt{16}}{2 \times 1}$

$x_{1,2}=\frac{6\pm4}{2}$

$x_{1}= \frac{6 + 4}{2} = \frac{10}{2} = 5$

$x_{2}= \frac{6 - 4}{2} = \frac{2}{2} = 1$

$d)9 {x}^{2} + 6x = 24$

$9 {x}^{2} + 6x - 24 = 0$

$a = 9$

$b = 6$

$c = - 24$

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

$\Delta = {6}^{2} - 4 \times 9 \times ( - 24)$

$\Delta = 36 + 864 = 900 > 0 = > \exists \: x_{1} \: \neq \: x_{2}$

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

$x_{1,2}=\frac{-6\pm\sqrt{900}}{2 \times 9}$

$x_{1,2}=\frac{-6\pm30}{18}$

$x_{1}=\frac{-6 + 30}{18} = \frac{24}{18} = \frac{4}{3}$

$x_{2}=\frac{-6 - 30}{18} = - \frac{36}{18} = - 2$

$e)4 {x}^{2} - 48 = 4x$

$4 {x}^{2} - 4x - 48 = 0$

$a = 4$

$b = - 4$

$c = - 48$

$\Delta = {( - 4)}^{2} - 4 \times 4 \times ( - 48)$

$\Delta = 16 + 768$

$\Delta = 784 > 0 = > \exists \: x_{1} \: \neq \: x_{2}$

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

$x_{1,2}=\frac{-(-4) \: \pm\sqrt{784}}{2\times4}$

$x_{1,2}=\frac{4\pm28}{8}$

$x_{1}=\frac{4 + 28}{8} = \frac{32}{8} = 4$

$x_{2}=\frac{4 - 28}{8} = - \frac{24}{8} = - 3$

$f)16 {x}^{2} + 8x = 80$

$16 {x}^{2} + 8x - 80 = 0$

$a = 16$

$b = 8$

$c = - 80$

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

$\Delta = {8}^{2} - 4 \times 16 \times ( - 80)$

$\Delta = 64 + 5120$

$\Delta = 5184 > 0 = > \exists \: x_{1} \: \neq \: x_{2}$

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

$x_{1,2}=\frac{-8\pm\sqrt{5184}}{2 \times 16}$

$x_{1,2}=\frac{-8\pm72}{32}$

$x_{1} = \frac{ - 8 + 72}{32} = \frac{64}{32} = 2$

$x_{2} = \frac{ - 8 - 72}{32} = - \frac{80}{32} = - \frac{10}{4} = - \frac{5}{2}$