a)

\(P=\dfrac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{2x+\sqrt{x}}{\sqrt{x}}+\dfrac{2\left(x-1\right)}{\sqrt{x}-1}\\ P=\sqrt{x}.\left(\sqrt{x}-1\right)-2\sqrt{x}-1+2\sqrt{x}+2\\ P=x-\sqrt{x}-2\sqrt{x}-1+2\sqrt{x}+2\\ P=x-\sqrt{x}+1\)

b)

\(P=\sqrt{x}^2-\sqrt{x}+1\\ P=\sqrt{x}^2-2.\dfrac{1}{2}\sqrt{x}+\dfrac{1}{4}+\dfrac{3}{4}\\ P=\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\)

dấu "=" xảy ra khi x=1/4

c)

vì \(2\sqrt{x}\) có bậc thấp hơn \(x-\sqrt{x}+1\) nên Q không thể là 1 số nguyên khác 0

TH nếu Q=0 thì x=0

vậy để Q nhận GT nguyên thì x=0

\(\sqrt{x+5}=1+\sqrt{x}\\ \Leftrightarrow x+5=1+2\sqrt{x}+x\\ \Leftrightarrow2\sqrt{x}=4\Rightarrow\sqrt{x}=2\Rightarrow x=4\)

1+5+9+13+17+...+x=501501

(x+1)[(x-1):4+1]:2=501501

(x+1)[(x-1).1/4+1].1/2=501501

(x+1)(1/4x-1/4+1).1/2=501501

[(x+1).1/4x-(x+1).1/4+(x+1)].1/2=501501

[(x+1)(-1/4+1)+(x+1).1/4x].1/2=501501

[(x+1).3/4+(x+1).1/4x].1/2=501501

(x+1)(3/4+x/4).1/2=501501

(x+1)[(3+x)/4] .1/2=501501

(x+1)[(3+x)/8] =501501

[(x+1)(3+x)]/8=501501

(x+1)(3+x)=4012008

3x+x^2+3+x=4012008

x^2+4x+3=4012008

x^2+x+3x+3=4012008

x(x+1)+3(x+1)=4012008

(x+1)(x+3)=4012008

(x+1)(x+3)=2002.2004

suy ra x+1=2002 và x+3=2004

           x=2001     ; x=2001

vậy x=2001

A)

\(x^3+8-4x^2-8x=\left(x+2\right)\left(x^2-2x+4\right)-4x\left(x+2\right)\\ \left(x+2\right)\left(x^2-6x+4\right)\)

B)

\(x^2+2x+1-y^2=\left(x+1\right)^2-y^2\\ =\left(x+1+y\right)\left(x+1-y\right)\)

C)

\(x^3-64=x^3-4^3\\ \left(x-4\right)\left(x^2+4x+16\right)\)

\(-\dfrac{1}{11}\)

\(x^{10}+x^8+1\\ =x^{10}+x^9+x^8-x^9-x^8-x^7++x^8+x^7+x^6-x^6-x^5-x^4+x^5+x^4+x^3-x^3-x^2-x+x^2+x+1\\ =x^8\left(x^2+x+1\right)-x^7\left(x^2+x+1\right)+x^6\left(x^2+x+1\right)-x^4\left(x^2+x+1\right)+x^3\left(x^2+x+1\right)-x\left(x^2+x+1\right)+1\left(x^2+x+1\right)\\ =\left(x^2+x+1\right)\left(x^8-x^7+x^6-x^4+x^3-x+1\right)\)

kẻ Bz//Cy//Ax

vì Bz//Cy nên zBC=BCy

ta có:

xAB+ABC-BCy=xAB+ABz+BCy-BCy=xAB+ABz=180

\(\dfrac{3}{4}-x=\dfrac{5}{9}+\dfrac{6}{11}\\ -x=\dfrac{139}{396}\Rightarrow x=-\dfrac{139}{396}\)

\(x:\dfrac{4}{28}=\dfrac{13}{-19}+\dfrac{8}{25}\\ 7x=-\dfrac{173}{475}\\ x=-\dfrac{\dfrac{173}{475}}{7}=-\dfrac{173}{3325}\)

\(y+30\%y=-1,3\\ 130\%y=-1,3\\ \Rightarrow y=\dfrac{-1,3}{130\%}=-1\)