А7+а9=12 а6*а10=-28 Найти а1 и d

Question

А7+а9=12
а6*а10=-28
Найти а1 и d

Answer 1

А7=а1+6d
a9=a1+8d
a6=a1+5d
a10=a1+9d
a1+6d+a1+8d=12
2a1+14d=12
a1+7d=6
a1=6–7d

(6–7d+5d)(6–7d+9d)=–28
(6–2d)(6+2d)=–28
36–4d^2=–28
–4d^2=–64
d^2=16
d=4; d=–4
а1=–22; d=34

При d=4; a1=–22:
а7=а1+6d=–22+24=2
a9=a1+8d=–22+32=10
a6=a1+5d=–22+20=–2
a10=a1+9d=–22+36=14

При d=–4; a1=34:
а7=а1+6d=34–24=10
a9=a1+8d=34–32=2
a6=a1+5d=34–20=14
a10=a1+9d=34–36=–2
Оба значения подходят к решению

Answer 2

а7=а1+6d

a9=a1+8d

a6=a1+5d

a10=a1+9d

составим систему :

a7+a9= а1+6d +a1+8d=12

a6*a10= (a1+5d)*(a1+9d)=-28

 

2a1+14d=12

a1^2+14a1d+45d^2=-28

 

a1+7d=6

a1^2+14 d a1+45d^2=-28

 

a1=6-7d

(6-7d)^2+14d(6-7d)+ 45d^2=-28

 

a1=6-7d

36-84d+49d^2+84d-98d^2+45d^2=-28

 

a1=6-7d

36-4d^2=-28

 

a1=6-7d

9- d^2=-7

 

a1=6-7d

d=±4

 

a1=6-7*4=-22

d=4

 

a1=6-7*(-4)=34

d=-4