Question

Equação do 2 grau.
A)3x(x+1)+(x+3)²=x+9
B)(x+2)²+(x+2)²=106
C)(x-1)²=3x+1

Answer 1

a) $3x(x+1)+(x+3)^2=x+9$
$3x^2+3x+x^2+6x+9=x+9$
$4x^2+9x+9=x+9$
$4x^2+8x=0$

Simplificando por $4$:

$\dfrac{4x}{4}+\dfrac{8x}{4}=\dfrac{0}{4}$
$x^2+2x=0$
$x\cdot(x+2)=0$

$\boxed{x'=0}$

$x+2=0~\longrightarrow~\boxed{x"=-2}$

$\text{S}=\{-2,0\}$


b) $(x+2)^2+(x+2)^2=106$
$x^2+4x+4+x^2+4x+4=106$
$2x^2+8x+8-106=0$
$2x^2+8x-98=0$

$x^2+4x-49=0$
$\Delta=4^2-4\cdot1\cdot(-49)$
$\Delta=16+196$
$\Delta=212$

$212=2^2\cdot53$

$\sqrt{212}=\sqrt{2^2\cdot53}=2\sqrt{53}$

$x=\dfrac{-4\pm\sqrt{212}}{2\cdot1}=\dfrac{-4\pm2\sqrt{53}}{2}=-2\pm\sqrt{53}$

$\boxed{x'=-2+\sqrt{53}}$

$\boxed{x"=-2-\sqrt{53}}$

$\text{S}=\{-2-\sqrt{53},-2+\sqrt{53}\}$


c) $(x-1)^2=3x+1$
$x^2-2x+1=3x+1$
$x^2-5x=0$
$x\cdot(x-5)=0$

$\boxed{x'=0}$

$x-5=0~\longrightarrow~\boxed{x"=5}$

$\text{S}=\{0,5\}$