a: \(x^4-2x^3-25x^2+50x=0\)
=>\(x^3\left(x-2\right)-25x\left(x-2\right)=0\)
=>\(\left(x-2\right)\left(x^3-25x\right)=0\)
=>x(x-2)(x^2-25)=0
=>x(x-2)(x+5)(x-5)=0
=>x∈{0;2;-5;5}
b: \(x^2\left(x-1\right)-4x^2+8x-4=0\)
=>\(x^2\left(x-1\right)-4\left(x^2-2x+1\right)=0\)
=>\(\left(x-1\right)\left(x^2-4x+4\right)=0\)
=>\(\left(x-1\right)\left(x-2\right)^2=0\)
=>x∈{1;2}
c: \(9x^2-4-2\left(3x-2\right)^2=0\)
=>(3x-2)(3x+2)-(3x-2)(6x-4)=0
=>(3x-2)(3x+2-6x+4)=0
=>(3x-2)(-3x+6)=0
=>(x-2)(3x-2)=0
=>x∈{2;2/3}
d: \(9x^2+90x+225-\left(x-7\right)^2=0\)
=>\(\left(3x+15\right)^2-\left(x-7\right)^2=0\)
=>(3x+15-x+7)(3x+15+x-7)=0
=>(2x+22)(4x+8)=0
=>2(x+11)*4*(x+2)=0
=>(x+11)(x+2)=0
=>x∈{-11;-2}
e: \(x^3-8+\left(x-2\right)\left(x+1\right)=0\)
=>\(\left(x-2\right)\left(x^2+2x+4\right)+\left(x-2\right)\left(x+1\right)=0\)
=>\(\left(x-2\right)\left(x^2+2x+4+x+1\right)=0\)
=>\(\left(x-2\right)\left(x^2+3x+5\right)=0\)
mà \(x^2+3x+5=x^2+3x+\frac94+\frac{11}{4}=\left(x+\frac32\right)^2+\frac{11}{4}>0\forall x\)
nên x-2=0
=>x=2
g: (x+1)(x+2)(x+3)(x+4)-24=0
=>\(\left(x^2+5x+4\right)\left(x^2+5x+6\right)-24=0\)
=>\(\left(x^2+5x\right)^2+10\left(x^2+5x\right)+24-24=0\)
=>\(\left(x^2+5x\right)\left(x^2+5x+10\right)=0\)
mà \(x^2+5x+10=x^2+5x+\frac{25}{4}+\frac{75}{4}=\left(x+\frac52\right)^2+\frac{75}{4}\ge\frac{75}{4}>0\forall x\)
nên \(x^2+5x=0\)
=>x(x+5)=0
=>x∈{0;-5}
