Question

三个有理数a.b.c.满足a:b:c=2:3:5，且a²+b²+c²=abc，则a+b+c= 。

要通俗易懂的过程

Answer 1

a:b:c=2:3:5
设:a=2R,b=3R,c=5R (R为正整数) {代入 a²+b²+c²=abc中}
得: (2R)²+(3R)²+(5R)²= 2R x 3R x 5R
4R²+9R²+25R² = 30R³
38R² = 30R³
19 = 15R
R =19/15

a+b+c= 2R +3R+5R

=R(2+3+5)
=10R
= 10 x 19/15 {代入R的值}
= 38/3

则a+b+c= 38/3

Answer 2

a:b:c=2:3:5

b=3a/2,c=5a/2
a²+b²+c²=abc
a²+9a²/4+25a²/4=15a³/4
38a²=15a³
a²(15a-38)=0

a=0或38/15
a=0时，b=0, c=0, a+b+c=0
a=38/15时，b=19/5, c=19/3, a+b+c= (38+19*3+19*5)/15=190/15=38/3

Answer 3

a:b:c=2:3:5

则令a=2x,b=3x,c=5x
所以4x²+9x²+25x²=30x³
x=19/15
所以a+b+c=10x=38/3

Answer 4

令abc为2k.3k.5k
则4k*2+9k*2+25k*2=30k*3
38k*2=30k*3
K=19/15
10k=38/3
则答案为38/3

Answer 5

a=2x
b=3x
c=5x
得38x^2=30x＾3
x=19/15
a+b+c=190/15