Mathematics Golden - Vol. 1 Ch. 4 - Door Feat. Inagawa Junji
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Solution to the equation:
x+√x(x+1)+√x(x+2)+√(x+1)(x+2)=2
The domain here is x≥0 or x≤-2. Here I'll only deal with the nonnegative case, for reasons I'll mention later.
We take note of the factorization
ac+ad+bc+bd=(a+b)(c+d)
Splitting the square roots and factorizing, we obtain
(√x+√(x+1))(√x+√(x+2))=2
Taking the reciprocal (inverse) of both sides and rationalizing the denominator, we rearrange to obtain
(√x-√(x+1))(√x-√(x+2))=1
We expand the expression on the left and obtain something very similar to the original equation:
x-√x(x+1)-√x(x+2)+√(x+1)(x+2)=1
By adding this to the original equation, we have
2x+2√(x+1)(x+2)=3
This is now a high school math problem. Move 2x to the right, squaring and solving the final equation, we obtain x=1/24.
The same technique doesn't quite work for the negative case due to a small technical issue regarding separating two negative numbers inside a square root. WolframAlpha returns a gargantuan output for this particular case, which hints that there is no simple way to solve this.
x+√x(x+1)+√x(x+2)+√(x+1)(x+2)=2
The domain here is x≥0 or x≤-2. Here I'll only deal with the nonnegative case, for reasons I'll mention later.
We take note of the factorization
ac+ad+bc+bd=(a+b)(c+d)
Splitting the square roots and factorizing, we obtain
(√x+√(x+1))(√x+√(x+2))=2
Taking the reciprocal (inverse) of both sides and rationalizing the denominator, we rearrange to obtain
(√x-√(x+1))(√x-√(x+2))=1
We expand the expression on the left and obtain something very similar to the original equation:
x-√x(x+1)-√x(x+2)+√(x+1)(x+2)=1
By adding this to the original equation, we have
2x+2√(x+1)(x+2)=3
This is now a high school math problem. Move 2x to the right, squaring and solving the final equation, we obtain x=1/24.
The same technique doesn't quite work for the negative case due to a small technical issue regarding separating two negative numbers inside a square root. WolframAlpha returns a gargantuan output for this particular case, which hints that there is no simple way to solve this.
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