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Solve \(\large \sqrt{2x-4} - \sqrt{x+5} = 1\)

 Jul 5, 2020
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Solve for x :
sqrt(2 x - 4) - sqrt(x + 5) = 1

(sqrt(2 x - 4) - sqrt(x + 5))^2 = 1 + 3 x - 2 sqrt(x + 5) sqrt(2 x - 4) = 1 + 3 x - 2 sqrt((x + 5) (2 x - 4)) = 1:
1 + 3 x - 2 sqrt((x + 5) (2 x - 4)) = 1

Subtract 3 x + 1 from both sides:
-2 sqrt((x + 5) (2 x - 4)) = -3 x

Raise both sides to the power of two:
4 (x + 5) (2 x - 4) = 9 x^2

Expand out terms of the left hand side:
8 x^2 + 24 x - 80 = 9 x^2

Subtract 9 x^2 from both sides:
-x^2 + 24 x - 80 = 0

The left hand side factors into a product with three terms:
-(x - 20) (x - 4) = 0

Multiply both sides by -1:
(x - 20) (x - 4) = 0

Split into two equations:
x - 20 = 0 or x - 4 = 0

Add 20 to both sides:
x = 20 or x - 4 = 0

Add 4 to both sides:
x = 20 or x = 4

sqrt(2 x - 4) - sqrt(x + 5) ⇒ sqrt(2 4 - 4) - sqrt(4 + 5) = -1:
So this solution is incorrect

sqrt(2 x - 4) - sqrt(x + 5) ⇒ sqrt(2 20 - 4) - sqrt(20 + 5) = 1:
So this solution is correct

The solution is:

 x = 20

 Jul 5, 2020

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