[tex]1)a) {x}^{2} = 1[/tex]
[tex]x = \pm \sqrt{1} [/tex]
[tex]x = \pm1[/tex]
[tex]b)2 {x}^{2} + 1 = - 7[/tex]
[tex]2 {x}^{2} = 7 + 1[/tex]
[tex]2 {x}^{2} = 8 \: | \div 2[/tex]
[tex] {x}^{2} = 4[/tex]
[tex]x = \pm \sqrt{4} [/tex]
[tex]x = \pm2[/tex]
[tex]c) {x}^{2} + 2 {x}^{2} = 4 {x}^{2} [/tex]
[tex] {x}^{2} + 2 {x}^{2} - 4 {x}^{2} = 0[/tex]
[tex] - {x}^{2} = 0[/tex]
[tex] {x}^{2} = 0[/tex]
[tex]x = 0[/tex]
[tex]2)34 + 15 {(x + 3)}^{2} = {( - 13)}^{2} [/tex]
[tex]34 + 15( {x}^{2} + 6x + 9) = 169[/tex]
[tex]34 + 15 {x}^{2} + 90x + 135 = 169[/tex]
[tex]15 {x}^{2} + 90x + 34 + 135 - 169 = 0[/tex]
[tex]15 {x}^{2} + 90x + 169 - 169 = 0[/tex]
[tex]15 {x}^{2} + 90x + 0 = 0[/tex]
[tex]15 {x}^{2} + 90x = 0[/tex]
[tex]a = 15[/tex]
[tex]b = 90[/tex]
[tex]c = 0[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {90}^{2} - 4 \times 15 \times 0[/tex]
[tex]\Delta = 8100>0=>\:\exists\:x_{1}\:\neq\:x_{2}[/tex]
[tex]x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-90\pm\sqrt{8100}}{2 \times 15}[/tex]
[tex]x_{1,2}=\frac{-90\pm90}{30}[/tex]
[tex]x_{1}=\frac{-90 + 90}{30}[/tex]
[tex]x_{1}=\frac{0}{30}[/tex]
[tex]x_{1}=0[/tex]
[tex]x_{2}=\frac{-90 - 90}{30}[/tex]
[tex]x_{2}=\frac{-180}{30}[/tex]
[tex]x_{2}= - 6[/tex]
[tex]3) {( 3 + x)}^{2} + {(3 - x)}^{2} = 36[/tex]
[tex]{[ 3+ ( - 3)]}^{2} + {[3 - ( - 3)]}^{2} = 36[/tex]
[tex] {(3 - 3)}^{2} + {(3 + 3)}^{2} = 36[/tex]
[tex] {0}^{2} + {6}^{2} = 36[/tex]
[tex]0 + 36 = 36[/tex]
[tex]36 = 36 \: (A)[/tex]
[tex]=>a=-3\:este\:solutie\:a\:ecuatiei[/tex]
[tex]4) {x}^{2} + {(x + 2)}^{2} + {(x + 4)}^{2} = 4(3x + 8)[/tex]
[tex] {x}^{2} + {x}^{2} + 4x + 4 + {x}^{2} + 8x + 16 = 12x + 32[/tex]
[tex] {x}^{2} + {x}^{2} + {x}^{2} + 4x + 8x - 12x = 32-16-4[/tex]
[tex]3 {x}^{2} = 12\:|\div3[/tex]
[tex] {x}^{2} = 4 [/tex]
[tex] x= \pm\:\sqrt{4} [/tex]
[tex]x = \pm\:2[/tex]
[tex]5) {(3x + 4)}^{2} = {(2x + 1)}^{2} [/tex]
[tex]9 {x}^{2} + 24x + 16 = 4 {x}^{2} + 4x + 1[/tex]
[tex]9 {x}^{2} - 4 {x}^{2} + 24x - 4x + 16 - 1 = 0[/tex]
[tex]5 {x}^{2} + 20x + 15 = 0[/tex]
[tex]a = 5[/tex]
[tex]b = 20[/tex]
[tex]c = 15[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {20}^{2} - 4 \times 5 \times 15[/tex]
[tex]\Delta = 400 - 300[/tex]
[tex]\Delta = 100 > 0 = > \exists \: x_{1} \: \neq \: x_{2}[/tex]
[tex]x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}=\frac{-20\pm\sqrt{100}}{2 \times 5}[/tex]
[tex]x_{1,2}=\frac{-20\pm10}{10}[/tex]
[tex]x_{1}=\frac{-20 + 10}{10}[/tex]
[tex]x_{1}=\frac{ - 10}{10}[/tex]
[tex]x_{1}= - 1[/tex]
[tex]x_{2}=\frac{ - 20 - 10}{10}[/tex]
[tex]x_{2}=\frac{ - 30}{10}[/tex]
[tex]x_{2}= - 3[/tex]
[tex]6) {x}^{2} + 6x + 8 = 6 + 4 \sqrt{3} [/tex]
[tex] {x}^{2} + 6x + 8 - 6 - 4 \sqrt{3} = 0[/tex]
[tex] {x}^{2} + 6x + 2 - 4 \sqrt{3} = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 6[/tex]
[tex]c = 2 - 4 \sqrt{3} [/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {6}^{2} - 4 \times 1 \times (2 - 4 \sqrt{3} )[/tex]
[tex]\Delta = 36 + 8 - 16 \sqrt{3} [/tex]
[tex]\Delta = 44 - 16 \sqrt{3}>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{44 - 16 \sqrt{3} }}{2 \times 1}[/tex]
[tex]x_{1,2}=\frac{-6\pm\sqrt{44 - 16 \sqrt{3} }}{2}[/tex]
[tex]x_{1}=\frac{-6 + \sqrt{44 - 16 \sqrt{3} }}{2}[/tex]
[tex]x_{2}=\frac{-6 - \sqrt{44 - 16 \sqrt{3} }}{2}[/tex]
