Bài 1: `x + y = 9; xy = 14` (*)
`a.`
`x + y = 9 ⇒ x = 9 - y`
Thay `9 - y` vào (*), ta được:
`(9 - y)y = 14`
`⇔9 - y^2 = 14`
`⇔9 - y^2 - 14 = 0`
`⇔(y - 7)(y - 2) = 0`
`⇔`\(\left[ \begin{array}{l}y-7=0\\y-2=0\end{array} \right.\)
`⇔`\(\left[ \begin{array}{l}y=7\\y=2\end{array} \right.\)
`⇒`\(\left[ \begin{array}{l}x=9-7=2\\x=0-2=7\end{array} \right.\)
`+) y = 7; x = 2 ⇒ x - y = 2 - 7 = -5`
`+) y = 2; x = 7 ⇒ x - y = 7 - 2 = 5`
Vậy `x - y = ±5`
`b. `
`(x + y)^2 = 9^2 = 81`
`⇔x^2 + y^2 + 2xy = 81`
`⇔x^2 + y^2 + 2*14 = 81`
`⇔x^2 + y^2 = 53`
Vậy `x^2 + y^2 = 53`
`c.`
`x^3 + y^3 `
`= (x + y)(x^2 - xy + y^2)`
`= 9 * (53 - 14) = 9 * 39 = 351`
`d. `
`x^5 + y^5`
`= (x^3 + y^3)(x^2 + y^2) - x^2y^2(x + y)`
`= 351* 53-14^2*9=16839`
Bài 2:
`a.`
`A=263^2+74*263+37^2`
`=263^2+2*37*263+37^2`
`=(263+37)^2`
`=300^2 = 90 000`
`b.`
`B = 136^2-92*136+46^2`
`=136^2-2*46*136+46^2`
`=(136-46)^2`
`=90^2=8100`
`c.`
`C=(258^2-242^2)/(254^2-246^2)=((258-242)(258+242))/((254-246)(254+246))=(16*100)/(8*100)=2`
`d.`
`D=1^2-2^2+3^2-4^2+...+49^2-50^2`
`⇒-D=50^2-49^2+48^2-47^2+...+2^2-1^2`
`=(50-49)(50+49)+(48-47)(48+47)+...+(2-1)(2+1)`
`=50+49+48+47+46+45+...+2+1`
`=((1+50)*50)/2=1275`
`⇒D=-1275`
`e.`
`(100^2+98^2+96^2+...+2^2)-(99^2+97^2+95^2+...+1^2)`
`=100^2+98^2+96^2+...+2^2-99^2-97^2-95^2-...-1^2`
`=(100^2-99^2)+(98^2-97^2)+...+(2^2-1^2)`
`=(100-99)(100+99)+(98-97)(98+97)+...+(2-1)(2+1)`
`=100+99+98+97+...+2+1`
`=((1+100)*100)/2=5050`
`f.`
`(63^2-47^2)/(215^2-105^2)=((63-47)(63+47))/((215-105)(215+105))=(16*110)/(110*320)=1/20`
`g.`
`(437^2-363^2)/(537^2-463^2)=((437-363)(437+363))/((537-463)(537+463))=(74*800)/(74*1000)=4/5`
Bài 3:
`a.`
`A = 26^2 - 24^2 = (26-24)(26+24)=2*50=100`
`B = 27^2-25^2=(27-25)(27+25)=2*52=104`
Vì `100<104` nên `A<B`
Vậy `A < B`
`b.`
`A=2021*2019=(2020+1)(2020-1)=2020^2-1`
`B=2020^2`
Vì `2020^2-1<2020^2` nên `A<B`
Vậy `A<B`
