# How do you solve #sqrt(4x+17) +sqrt(x +7) =sqrt(x + 2)#?

##### 1 Answer

#### Answer:

#### Explanation:

Start by writing down the conditions that

#4x + 17>=0 implies x>= -17/4# #x+7 >=0 implies x>= -7# #x+2 >=0 implies x>= -2#

These conditions are determined by the fact that the expressions under the radicals **must be positive** if you're dealing with real numbers.

Combine all three conditions to get

Another important thing to notice here is that for any

#4x+17 >x+7#

Moreover,

#x+7 > x+2#

This actually tells you that the equation has **no valid solutions** among real numbers, since **two positive numbers** cannot be added to give a **smaller** positive number.

In other simply, any solution that you will come about by solving this equation will be *extraneous*.

Now, square both sides of the equation to reduce the number of radical terms from three to one.

#(sqrt(4x+17) + sqrt(x+7))^2 = (sqrt(x+2))^2#

#(sqrt(4x+17))^2 + 2sqrt((4x+17)(x+7)) + (sqrt(x+7))^2 = x+2#

#4x+17 + 2sqrt((4x+17)(x+7)) + color(red)(cancel(color(black)(x))) + 7= color(red)(cancel(color(black)(x))) + 2#

This is equivalent to

#2sqrt((4x+17)(x+7)) = -4x - 22#

Square both sides of the equation again to get rid of the last radical term

#(2sqrt((4x+17)(x+7)))^2 = (-4x - 22)""^2#

#4(4x+17)(x+7) = 16x^2 + 176x + 484#

#color(red)(cancel(color(black)(16x^2))) + 180x + 476 = color(red)(cancel(color(black)(16x^2))) + 176x + 484#

This is equivalent to

#4x = 8 implies x = 8/4 = color(green)(2)#

Notice that **not** a valid solution

#sqrt(4 * (2) + 17) + sqrt(2 + 7) = sqrt(2 + 2)#

#sqrt(25) + sqrt(9) = sqrt(4)#

#5 + 3 color(Red)(!=) 2 -> x = 2# is anextraneous solution.