[tex]a)(x + 2)(x - 5) = 0[/tex]
[tex] = > x + 2 = 0 = > x_{1} = - 2[/tex]
[tex] = > x - 5 = 0 = > x_{2} = 5[/tex]
[tex]b) {x}^{2} - 4 = 0[/tex]
[tex] {x}^{2} = 4[/tex]
[tex]x = \pm \sqrt{4} [/tex]
[tex]x = \pm2[/tex]
[tex] = > x_{1} = 2[/tex]
[tex] = > x_{2} = - 2[/tex]
[tex]c) {(x + 2)}^{2} = 5(x + 2)[/tex]
[tex] {x}^{2} + 4x + 4 = 5x + 10[/tex]
[tex] {x}^{2} + 4x - 5x + 4 - 10 = 0[/tex]
[tex] {x}^{2} - x - 6 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = - 1[/tex]
[tex]c = - 6[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {( - 1)}^{2} - 4 \times 1 \times ( - 6)[/tex]
[tex]\Delta = 1 + 24[/tex]
[tex]\Delta = 25>0=>\exists\:x_{1}\:\neq\:x_{2}\:\epsilon\: \mathbb{R}[/tex]
[tex]x_{1,2} = \frac{ - b \pm \sqrt{\Delta} }{2a} [/tex]
[tex]x_{1,2} = \frac{ - ( - 1) \pm \sqrt{25} }{2 \times 1} [/tex]
[tex]x_{1,2} = \frac{1 \pm5}{2} [/tex]
[tex]x_{1} = \frac{1 + 5}{2} = \frac{6}{2} = 3[/tex]
[tex]x_{2} = \frac{1 - 5}{2} = \frac{ - 4}{2} = - \frac{4}{2} = - 2[/tex]
[tex]e) {x}^{2} - 3x = 0[/tex]
[tex]a = 1[/tex]
[tex]b = - 3[/tex]
[tex]c = 0[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {( - 3)}^{2} - 4 \times 1 \times 0[/tex]
[tex]\Delta = 9>0=>\exists\:x_{1}\:\neq\:x_{2}\:\epsilon\: \mathbb{Q}[/tex]
[tex]x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}[/tex]
[tex]x_{1,2}= \frac{ - ( - 3) \pm \sqrt{9} }{2 \times 1} [/tex]
[tex]x_{1,2}=\frac{3\pm3}{2}[/tex]
[tex]x_{1}=\frac{3 + 3}{2} = \frac{6}{2} = 3[/tex]
[tex]x_{2}=\frac{3 - 3}{2} = \frac{0}{2} = 0[/tex]
[tex]g)2 {x}^{2} - 1 = 3 {x}^{2} + 1[/tex]
[tex]2 {x}^{2} - 3 {x}^{2} = 1 + 1[/tex]
[tex] - {x}^{2} = 2 \: | \times ( - 1)[/tex]
[tex] {x}^{2} = - 2[/tex]
[tex]x = \pm \sqrt{ - 2} = > nu \: exista[/tex]
