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sqrt(x-2)+sqrt(x+1)= 27 sqrt(x+1)-sqrt(x-2)=?

Guest Mar 17, 2017
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 #1
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Solve for x:
sqrt(x - 2) + sqrt(x + 1) = 27 sqrt(x + 1) - sqrt(x - 2)

(sqrt(x - 2) + sqrt(x + 1))^2 = -1 + 2 x + 2 sqrt(x - 2) sqrt(x + 1) = -1 + 2 x + 2 sqrt((x - 2) (x + 1)) = (27 sqrt(x + 1) - sqrt(x - 2))^2:
-1 + 2 x + 2 sqrt((x - 2) (x + 1)) = (27 sqrt(x + 1) - sqrt(x - 2))^2

(27 sqrt(x + 1) - sqrt(x - 2))^2 = 727 + 730 x - 54 sqrt(x - 2) sqrt(x + 1):
-1 + 2 x + 2 sqrt((x - 2) (x + 1)) = 727 + 730 x - 54 sqrt(x - 2) sqrt(x + 1)

Subtract 2 x - 1 from both sides:
2 sqrt((x - 2) (x + 1)) = 728 + 728 x - 54 sqrt(x - 2) sqrt(x + 1)

Raise both sides to the power of two:
4 (x - 2) (x + 1) = (728 + 728 x - 54 sqrt(x - 2) sqrt(x + 1))^2

Combine sqrt(x - 2) and sqrt(x + 1) under the same square root:
4 (x - 2) (x + 1) = (728 + 728 x - 54 sqrt((x - 2) (x + 1)))^2

Expand out terms of the left hand side:
4 x^2 - 4 x - 8 = (728 + 728 x - 54 sqrt((x - 2) (x + 1)))^2

(728 + 728 x - 54 sqrt((x - 2) (x + 1)))^2 = 524152 + 1057052 x + 532900 x^2 + (-78624 - 78624 x) sqrt((x - 2) (x + 1)):
4 x^2 - 4 x - 8 = 524152 + 1057052 x + 532900 x^2 + sqrt((x - 2) (x + 1)) (-78624 - 78624 x)

Subtract 524152 + 1057052 x + 532900 x^2 + (-78624 - 78624 x) sqrt((x - 2) (x + 1)) from both sides:
-524160 - 1057056 x - 532896 x^2 - sqrt((x - 2) (x + 1)) (-78624 - 78624 x) = 0

Add 532896 x^2 + 1057056 x + 524160 to both sides:
sqrt((x - 2) (x + 1)) (78624 x + 78624) = 532896 x^2 + 1057056 x + 524160

Raise both sides to the power of two:
(x - 2) (x + 1) (78624 x + 78624)^2 = (532896 x^2 + 1057056 x + 524160)^2

Expand out terms of the left hand side:
6181733376 x^4 + 6181733376 x^3 - 18545200128 x^2 - 30908666880 x - 12363466752 = (532896 x^2 + 1057056 x + 524160)^2

Expand out terms of the right hand side:
6181733376 x^4 + 6181733376 x^3 - 18545200128 x^2 - 30908666880 x - 12363466752 = 283978146816 x^4 + 1126601828352 x^3 + 1676012921856 x^2 + 1108132945920 x + 274743705600

Subtract 283978146816 x^4 + 1126601828352 x^3 + 1676012921856 x^2 + 1108132945920 x + 274743705600 from both sides:
-277796413440 x^4 - 1120420094976 x^3 - 1694558121984 x^2 - 1139041612800 x - 287107172352 = 0

The left hand side factors into a product with four terms:
-76317696 (x + 1)^2 (56 x + 57) (65 x + 66) = 0

Divide both sides by -76317696:
(x + 1)^2 (56 x + 57) (65 x + 66) = 0

Split into three equations:
(x + 1)^2 = 0 or 56 x + 57 = 0 or 65 x + 66 = 0

Take the square root of both sides:
x + 1 = 0 or 56 x + 57 = 0 or 65 x + 66 = 0

Subtract 1 from both sides:
x = -1 or 56 x + 57 = 0 or 65 x + 66 = 0

Subtract 57 from both sides:
x = -1 or 56 x = -57 or 65 x + 66 = 0

Divide both sides by 56:
x = -1 or x = -57/56 or 65 x + 66 = 0

Subtract 66 from both sides:
x = -1 or x = -57/56 or 65 x = -66

Divide both sides by 65:
x = -1 or x = -57/56 or x = -66/65

sqrt(x - 2) + sqrt(x + 1) ⇒ sqrt(-2 - 57/56) + sqrt(1 - 57/56) = i sqrt(7/2) ≈ 1.87083 i
27 sqrt(x + 1) - sqrt(x - 2) ⇒ 27 sqrt(1 - 57/56) - sqrt(-2 - 57/56) = i sqrt(7/2) ≈ 1.87083 i:
So this solution is correct

sqrt(x - 2) + sqrt(x + 1) ⇒ sqrt(-2 - 66/65) + sqrt(1 - 66/65) = 3 i sqrt(5/13) ≈ 1.86052 i
27 sqrt(x + 1) - sqrt(x - 2) ⇒ 27 sqrt(1 - 66/65) - sqrt(-2 - 66/65) = i sqrt(13/5) ≈ 1.61245 i:
So this solution is incorrect

sqrt(x - 2) + sqrt(x + 1) ⇒ sqrt(-2 - 1) + sqrt(1 - 1) = i sqrt(3) ≈ 1.73205 i
27 sqrt(x + 1) - sqrt(x - 2) ⇒ 27 sqrt(1 - 1) - sqrt(-2 - 1) = -i sqrt(3) ≈ -1.73205 i:
So this solution is incorrect

The solution is:
Answer: |x = -57/56 - (assuming a complex-valued square root)

Guest Mar 17, 2017
 #2
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Probably better to add sqrt(x-2) to both sides and subtract sqrt(x+1) from both sides first, before thinking about squaring!

Alan  Mar 17, 2017

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