Solution: x≠-1/2, -1 original equation becomes (4x+3)²=18/(2x+1) (2x+2) Introduce parameters a, ba=2x+1b=2x+2a+b=4x+3∴(a+b)²=18/ab…①∴ab(a+b)²=18…② Also ∵b-a=1, ∴b=a+1…③ Substitute ②a(a+1) (2a+1)²=18 (a²+a) (4a

2025/04/2808:19:33 education 1689

Solution: x≠-1/2, -1

The original equation becomes

(4x+3)²=18/(2x+1) (2x+2)

Introduction parameters a, b

a=2x+1

b=2x+2

a+b=4x+3

∴(a+b)²=18/ab…①

∴ab(a+b)²=18…②

3 Also ∵b-a=1, ∴b=a+1…③ Substitute ②

a(a+1)(2a+1)²=18

(a²+a)(4a²+4a+1)=18

∴(4a²+4a)(4a²+4a+1)=72

(4a²+4a)²+(4a²+4a)-72=0

∴(4a²+4a+9)(4a²+4a-8)=0

∴(4a²+4a+9)(a²+a-2)=0

∴(4a²+4a+9)(a+2)(a-1)=0

∴a=-2 or a=1

When a=-2, 2x+1=-2, x=-3/2

When a=1, 2x+1=1, x=0

∴The solution of the original equation is: x1=-3/2, x2=0

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