Question

A) (2x + 3) (x - 2) = (x +1 ) (x - 6)

B) x² = 12x - 11

C) 2 (x - 1)² = x

D)(x - 2)² = x + 1

E) (3x - 5) (2x - 5) = (x + 3) (x - 1)

F) x² - 1 = x - 1 dividido por 4

G) 2x (x + 5) = x² + 5x + 14

Answer 1

$A) ~~(2x + 3) (x - 2) = (x +1 ) (x - 6) \to\\\\ 2x^2-4x+3x-6=x^2-6x+x-6\to\\\\ 2x^2-x-6=x^2-5x-6\to\\ \\2x^2-x-\not6-x^2+5x+\not6=0\to\\\\ x^2+4x=0\to~~~~Por~~Evid\^encia:\\\\\\ x(x+4)=0\to\\\\ \boxed{x'=0}\\\\x+4=0\to~~\boxed{x''=-4} \\\\\\\\ \large\boxed{\boxed{S=\{-4~;~0\}}}$





$B) ~~x^2 = 12x - 11\to\\ \\ x^{2} -12x+11=0\to~~~~Fatorando:\\\\(x-11)(x-1)=0\\\\x-11=0\to~~ \boxed{x'=11} \\\\ x-1=0\to~~ \boxed{x''=1} \\\\\\ \large\boxed{\boxed{S=\{1~;~11\}}}$





$C)~~ 2 (x - 1)^2 = x \to\\\\ 2 (x - 1)(x-1) = x \to\\\\ 2(x^2-x-x+1)=x\to\\\\2(x^2-2x+1)=x\to\\\\ 2x^2-4x+2-x=0\to\\\\\\ 2x^2-5x+2=0\\\\ a=2~;~b=-5~;~c=2\\\\\\ \Delta=b^2-4ac\to \Delta=(-5)^2-4.2.2\to \Delta=25-16\to \boxed{\Delta=9} \\\\x' \neq x''$

$x= \dfrac{-b\pm \sqrt{\Delta} }{2a} \to~~ x= \dfrac{-(-5)\pm \sqrt{9} }{2.2} \to~~ x= \dfrac{5\pm 3 }{4} \to\\\\\\ x'= \dfrac{5+ 3 }{4} \to~~ x'= \dfrac{8}{4} \to~~ \boxed{x'=2}\\\\\\ x''= \dfrac{5- 3 }{4} \to~~ x''= \dfrac{2}{4} \to~~ \boxed{x''= \dfrac{1}{2} }\\\\\\\\ \boxed{\boxed{S= \left\{ \dfrac{1}{2} ~;~2\right\} }}$





$D)~~(x - 2)^2 = x + 1 \to\\\\ (x - 2)(x - 2) = x + 1 \to\\\\ x^2-2x-2x+4=x+1\to\\\\ x^2-4x+4-x-1=0\to\\\\\\ x^2 -5x + 3 = 0\to\\\\ a=1~;~b=-5~;~c=3\\\\\\ \Delta=(-5)^2-4.1.3\to \Delta=25-12\to \boxed{\Delta= 13} \\\\x' \neq x''$

$x= \dfrac{(-5)\pm \sqrt{13} }{2.1} \to ~~ x= \dfrac{5\pm \sqrt{13} }{2} \to\\\\\\ x ' = \dfrac{5+ \sqrt{13} }{2} ~~~~e~~~~ x'' = \dfrac{5- \sqrt{13} }{2} \\\\\\\\ \large\boxed{\boxed{ S=\left\{ \dfrac{5- \sqrt{13} }{2} ~;~ \dfrac{5- \sqrt{13} }{2} \right\}}}$

Answer 2

 Anamary,
Complementando o bom trabalho da Luana

E)
   Efetuandfo
         6x^2 - 25x + 25 = x^2 + 2x - 4

  Reduzindo termos semelhantes
         5x^2 - 27x + 30 = 0

 Pela fórmula resolutiva
         x = [27 +/- √[(27^2 - 4(5)(30)]/(2.5)
            = [27+/- √(729 - 600)]/10
            = (27 +/- √129)/10

                       x1 = (27 - √1290/10
                       x2 = (27 + √129)/10
                                                     S = (27 - √129)/10, (27 + √129)/10}

Igual anterior
F)
         x^2 - 1 = (x - 1)/4

       4x^2 - 4 = x - 1 = 0
       4x^2 - x - 3 = 0

      Fatorando
       (x - 1)(4x + 3) = 0
               x - 1 = 0
                                 x1 = 1
             4x + 3 = 0
               4x = - 3
                                 x2 = - 3/4
                                                 S = {- 3/4, 1 }

G)
           2x^2 + 10x = x^2 + 5x + 14
           x^2 + 5x - 14 = 0

      Fatorando  
           (x - 2)(x + 7)
                   x - 2 = 0
                                     x1 = 2
                   x + 7 = 0
                                     x2 = - 7
                                                           S = { - 7, 2 }