Answer
A) $x=6$
B)$x=1$
C) $x-4)(x+3)$
D) $-\frac{\sqrt 2}{2}$ and $-\frac{3\sqrt 2}{2}$
E)$[\sqrt 2]=2$
NO SOLUTION
F) $(x^{2}-2)(x+1)(x-1)$
G) $x=\frac{22}{3}$. And. $x=\frac{2}{3}$
Work Step by Step
A) $x+5=14-\frac{1}{2}x$
$(x+\frac{1}{2}x)=14-5$
$\frac{3}{2}x=9$
$x=9\div\frac{3}{2}$
$x=6$
B)$\frac{2x}{x+1}=\frac{2x-1}{x}$
Cross multiply
$2x^{2}=2x^{2}+x-1$
$2x^{2}-2x^{2}+x-1=0$
$x-1=0$
$x=1$
C) $x^{2}-x-12=0$
$x-4)(x+3)$
D) $2x^{2}+4x+1=0$
Quadratic Formula $\frac{-b\sqrt (b^{2}-4ac)}{2a}$
$\frac{-4\sqrt (4^{2}-4(2)(1))}{2(2)}$
$\frac{-4\sqrt (8)}{4}$
$\frac{-4\sqrt (4\times2)}{4}$
$\frac{-4+2sqrt 2}{4}$. $\frac{-4-2\sqrt 2}{4}$
$\frac{-2\sqrt 2}{4}$. $\frac{-6\sqrt 2}{4}$
$-\frac{\sqrt 2}{2}$ and $-\frac{3\sqrt 2}{2}$
E)$[\sqrt 3-\sqrt (x+5)]=2$
Square both sides $[\sqrt 3^{2}-\sqrt (x+5)]=2^{2}$
$3-\sqrt (x+5)]=4$
$-\sqrt (x+5)]=1$
$\sqrt (x+5)=-1$
Square both sides. $\sqrt (x+5)^{2}]=-1^{2}$
$(x+5)=1$
$x=-4$
Check for extraneous root
$[\sqrt 3-\sqrt (-4+5)]=2$
$[\sqrt 3-\sqrt (1)]=2$
$[\sqrt 3-1]=2$
$[\sqrt 2]=2$
NO SOLUTION
F) x^{4}-3x^{2}+2=0
$(x^{2}-2)(x+1)(x-1)$
G) $3|x-4|=10$
$|x-4|=\frac{10}{3}$
$(x-4)= \frac{10}{3}$. And. $(x-4)=\frac-{10}{3}
$x=\frac{10}{3}+4$. and. $x=\frac-{10}{3}+4$
$x=\frac{22}{3}$. And. $x=\frac{2}{3}$
