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# hard algebra

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Solve $$x-\sqrt{\frac{x}{2}+\frac{7}{8}-\sqrt{\frac{x}{8}+\frac{13}{64}}}=179$$

Jul 7, 2020

#1
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Solve for x:
x - sqrt(7/8 - sqrt(x/8 + 13/64) + x/2) = 179

Rewrite the left hand side by combining fractions. x - sqrt(7/8 - sqrt(x/8 + 13/64) + x/2) = 1/4 (4 x - sqrt(2) sqrt(7 + 4 x - sqrt(8 x + 13))):
1/4 (4 x - sqrt(2) sqrt(7 + 4 x - sqrt(8 x + 13))) = 179

Multiply both sides by 4:
4 x - sqrt(2) sqrt(7 + 4 x - sqrt(8 x + 13)) = 716

Subtract 4 x from both sides:
-sqrt(2) sqrt(7 + 4 x - sqrt(8 x + 13)) = 716 - 4 x

Raise both sides to the power of two:
2 (7 + 4 x - sqrt(8 x + 13)) = (716 - 4 x)^2

2 (7 + 4 x - sqrt(8 x + 13)) = 14 + 8 x - 2 sqrt(8 x + 13):
14 + 8 x - 2 sqrt(8 x + 13) = (716 - 4 x)^2

Subtract 8 x + 14 from both sides:
-2 sqrt(8 x + 13) = -14 + (716 - 4 x)^2 - 8 x

Raise both sides to the power of two:
4 (8 x + 13) = (-14 + (716 - 4 x)^2 - 8 x)^2

Expand out terms of the left hand side:
32 x + 52 = (-14 + (716 - 4 x)^2 - 8 x)^2

Expand out terms of the right hand side:
32 x + 52 = 256 x^4 - 183552 x^3 + 49306240 x^2 - 5881029024 x + 262801820164

Subtract 256 x^4 - 183552 x^3 + 49306240 x^2 - 5881029024 x + 262801820164 from both sides:
-256 x^4 + 183552 x^3 - 49306240 x^2 + 5881029056 x - 262801820112 = 0

The left hand side factors into a product with four terms:
-16 (2 x - 377) (2 x - 339) (4 x^2 - 1436 x + 128519) = 0

Divide both sides by -16:
(2 x - 377) (2 x - 339) (4 x^2 - 1436 x + 128519) = 0

Split into three equations:
2 x - 377 = 0 or 2 x - 339 = 0 or 4 x^2 - 1436 x + 128519 = 0

2 x = 377 or 2 x - 339 = 0 or 4 x^2 - 1436 x + 128519 = 0

Divide both sides by 2:
x = 377/2 or 2 x - 339 = 0 or 4 x^2 - 1436 x + 128519 = 0

x = 377/2 or 2 x = 339 or 4 x^2 - 1436 x + 128519 = 0

Divide both sides by 2:
x = 377/2 or x = 339/2 or 4 x^2 - 1436 x + 128519 = 0

Divide both sides by 4:
x = 377/2 or x = 339/2 or x^2 - 359 x + 128519/4 = 0

Subtract 128519/4 from both sides:
x = 377/2 or x = 339/2 or x^2 - 359 x = -128519/4

x = 377/2 or x = 339/2 or x^2 - 359 x + 128881/4 = 181/2

Write the left hand side as a square:
x = 377/2 or x = 339/2 or (x - 359/2)^2 = 181/2

Take the square root of both sides:
x = 377/2 or x = 339/2 or x - 359/2 = sqrt(181/2) or x - 359/2 = -sqrt(181/2)

x = 377/2 or x = 339/2 or x = 359/2 + sqrt(181/2) or x - 359/2 = -sqrt(181/2)

x = 377/2 or x = 339/2 or x = 359/2 + sqrt(181/2) or x = 359/2 - sqrt(181/2)

x - sqrt(7/8 - sqrt(x/8 + 13/64) + x/2) ⇒ 339/2 - sqrt(7/8 - sqrt(13/64 + 339/(8 2)) + 1/2×339/2) = 321/2:
So this solution is incorrect

x - sqrt(7/8 - sqrt(x/8 + 13/64) + x/2) ⇒ 377/2 - sqrt(7/8 - sqrt(13/64 + 377/(8 2)) + 1/2×377/2) = 179:
So this solution is correct

x - sqrt(7/8 - sqrt(x/8 + 13/64) + x/2) ≈ 160.974:
So this solution is incorrect

x - sqrt(7/8 - sqrt(x/8 + 13/64) + x/2) ≈ 179.5:
So this solution is incorrect

The solution is:

x = 377/2

Jul 7, 2020
#2
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Here's an alternative approach: Jul 7, 2020