+0

# Solve for x

+3
1161
3

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

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.....   Jan 16, 2016

#1
+5

That problem should be illegal...

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

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:

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