\(\sqrt\frac{5}{x+3}=\frac{4}{x+\sqrt4}\)

Please show how you got to your answer.

gibsonj338
Jan 16, 2016

#3**+15 **

sqrt [(5) / (x + 3)] = 4 / [x + sqrt(4)]

sqrt [ (5) / (x + 3)] = 4/ [x + 2] square both sides

5 / [x + 3] = 16/ [x^2 + 4x + 4] cross-multiply

5 [ x^2 + 4x + 4 ] = 16[x + 3] simplify

5x^2 + 20x + 20 = 16x + 48

5x^2 + 4x -28 = 0 factor

[5x + 14] [ x - 2] = 0

Setting each factor to O we have that x = -14/5 or x = 2

However.....x = -14/5 produces a negative quantity on the right hand side of the original equation, and we can never get a negative out of the left hand side......so.......the two sides would have unequal signs, so x = -14/5 is not a soluition

So.......x = 2 is the only solution.....

CPhill
Jan 16, 2016

#2**+10 **

Solve for x:

sqrt(5) sqrt(1/(x+3)) = 4/(x+2)

Cross multiply:

sqrt(5) sqrt(1/(x+3)) (x+2) = 4

Divide both sides by sqrt(5):

sqrt(1/(x+3)) (x+2) = 4/sqrt(5)

Divide both sides by x+2:

sqrt(1/(x+3)) = 4/(sqrt(5) (x+2))

Raise both sides to the power of two:

1/(x+3) = 16/(5 (x+2)^2)

Take the reciprocal of both sides:

x+3 = 5/16 (x+2)^2

Expand out terms of the right hand side:

x+3 = (5 x^2)/16+(5 x)/4+5/4

Subtract (5 x^2)/16+(5 x)/4+5/4 from both sides:

-(5 x^2)/16-x/4+7/4 = 0

The left hand side factors into a product with four terms:

-1/16 ((x-2) (5 x+14)) = 0

Multiply both sides by -16:

(x-2) (5 x+14) = 0

Split into two equations:

x-2 = 0 or 5 x+14 = 0

Add 2 to both sides:

x = 2 or 5 x+14 = 0

Subtract 14 from both sides:

x = 2 or 5 x = -14

Divide both sides by 5:

x = 2 or x = -14/5

sqrt(5) sqrt(1/(x+3)) => sqrt(5) sqrt(1/(3-14/5)) = 5

4/(x+2) => 4/(2-14/5) = -5:

So this solution is incorrect

sqrt(5) sqrt(1/(x+3)) => sqrt(5) sqrt(1/(3+2)) = 1

4/(x+2) => 4/(2+2) = 1:

So this solution is correct

The solution is:

**Answer: | x = 2**

Guest Jan 16, 2016

#3**+15 **

Best Answer

sqrt [(5) / (x + 3)] = 4 / [x + sqrt(4)]

sqrt [ (5) / (x + 3)] = 4/ [x + 2] square both sides

5 / [x + 3] = 16/ [x^2 + 4x + 4] cross-multiply

5 [ x^2 + 4x + 4 ] = 16[x + 3] simplify

5x^2 + 20x + 20 = 16x + 48

5x^2 + 4x -28 = 0 factor

[5x + 14] [ x - 2] = 0

Setting each factor to O we have that x = -14/5 or x = 2

However.....x = -14/5 produces a negative quantity on the right hand side of the original equation, and we can never get a negative out of the left hand side......so.......the two sides would have unequal signs, so x = -14/5 is not a soluition

So.......x = 2 is the only solution.....

CPhill
Jan 16, 2016