giải phương trình (x-2)(x-5)(x-6)(x-9)=(x+2)(x+5)(x+6)(x+9) 16/07/2021 Bởi Allison giải phương trình (x-2)(x-5)(x-6)(x-9)=(x+2)(x+5)(x+6)(x+9)
`(x-2)(x-5)(x-6)(x-9)=(x+2)(x+5)(x+6)(x+9)` `<=>x^4-9x^3-6x^3+54x^2-5x^3+45x^2+3x^2-270x-2x^3+18x^2+12x^2-108x+10x^2-90x-60x+540=x^4+9x^3+6x^3+54x^2+5x^3+45x^2+30x^2+270x+2x^3+18x^2+12x^2+108x+10x^2+90x+60x+540` `<=>22x^3-528x+22x^3+528x=0` `<=>44x^3+1056x=0` `<=>44x(x^2+24)=0` `<=>x=0` Vậy ……… Bình luận
$(x-2)(x-5)(x-6)(x-9)=(x+2)(x+5)(x+6)(x+9)$ $⇔(x^2-11x+18)(x^2-11x+30)=(x^2+11x+18)(x^2+11x+30)$ Đặt $x^2-11x+18=u,x^2+11x+18=v$ $⇒u(u+12)=v(v+12)$ $⇔u^2+12u-v^2-12v=0$ $⇔(u+v)(u-v)+12(u-v)=0$ $⇔(u-v)(u+v+12)=0$ $⇔-22x(2x^2+48)=0$ $⇔-44x(x^2+24)=0$ Vì $x^2≥0∀x⇒x^2+24>0∀x$ $⇔x=0$ Vậy $S=\{0\}$. Bình luận
`(x-2)(x-5)(x-6)(x-9)=(x+2)(x+5)(x+6)(x+9)`
`<=>x^4-9x^3-6x^3+54x^2-5x^3+45x^2+3x^2-270x-2x^3+18x^2+12x^2-108x+10x^2-90x-60x+540=x^4+9x^3+6x^3+54x^2+5x^3+45x^2+30x^2+270x+2x^3+18x^2+12x^2+108x+10x^2+90x+60x+540`
`<=>22x^3-528x+22x^3+528x=0`
`<=>44x^3+1056x=0`
`<=>44x(x^2+24)=0`
`<=>x=0`
Vậy ………
$(x-2)(x-5)(x-6)(x-9)=(x+2)(x+5)(x+6)(x+9)$
$⇔(x^2-11x+18)(x^2-11x+30)=(x^2+11x+18)(x^2+11x+30)$
Đặt $x^2-11x+18=u,x^2+11x+18=v$
$⇒u(u+12)=v(v+12)$
$⇔u^2+12u-v^2-12v=0$
$⇔(u+v)(u-v)+12(u-v)=0$
$⇔(u-v)(u+v+12)=0$
$⇔-22x(2x^2+48)=0$
$⇔-44x(x^2+24)=0$
Vì $x^2≥0∀x⇒x^2+24>0∀x$
$⇔x=0$
Vậy $S=\{0\}$.