How do you solve #sqrt(x+4 )= 1- sqrt(3x+13)#?

Question

Nathan Axel · Accepted Answer

#x=-4#

Given:

#sqrt(x+4) = 1-sqrt(3x+13)#

We can square both sides to get:

#x+4=1-2sqrt(3x+13)+(3x+13) = 3x+14-2sqrt(3x+13)#

Add #2sqrt(3x+13)-x-4# to both ends to get:

#2sqrt(3x+13) = 2x+10#

Divide both sides by #2# to get:

#sqrt(3x+13) = x+5#

Square both sides to get:

#3x+13 = x^2+10x+25#

Subtract #3x+13# from both sides to get:

#0 = x^2+7x+12 = (x+3)(x+4)#

So:

#x = -3" "# or #x = -4#

We now need to check for extraneous roots, since these may have been introduced when we squared the equation:

#sqrt((color(blue)(-3))+4) = 1 != -1 = 1-2 = 1-sqrt(4) = 1-sqrt(3(color(blue)(-3))+4)#

#sqrt((color(blue)(-4))+4) = 0 = 1-1 = 1-sqrt(1) = 1-sqrt(3(color(blue)(-4))+13)#

So #x=-3# is an extraneous solution caused by squaring #1 != -1#, while #x=-4# is a solution of the original equation.

Eva Adamson · Answer

undefined To solve the equation sqrt(x+4) = 1 - sqrt(3x+13), we can follow these steps:

1. Start by isolating one of the square roots on one side of the equation. In this case, let's isolate sqrt(x+4) by adding sqrt(3x+13) to both sides:
   sqrt(x+4) + sqrt(3x+13) = 1

2. Next, square both sides of the equation to eliminate the square roots:
   (sqrt(x+4) + sqrt(3x+13))^2 = 1^2
   (x+4) + 2sqrt(x+4)sqrt(3x+13) + (3x+13) = 1

3. Simplify the equation:
   x + 4 + 2sqrt(x+4)sqrt(3x+13) + 3x + 13 = 1
   4x + 17 + 2sqrt(x+4)sqrt(3x+13) = 1

4. Move all terms to one side of the equation:
   4x + 2sqrt(x+4)sqrt(3x+13) = 1 - 17
   4x + 2sqrt(x+4)sqrt(3x+13) = -16

5. Simplify the equation further:
   4x + 2sqrt(x+4)sqrt(3x+13) = -16

6. Square both sides of the equation again to eliminate the remaining square root:
   (4x + 2sqrt(x+4)sqrt(3x+13))^2 = (-16)^2
   16x^2 + 16sqrt(x+4)sqrt(3x+13) + 12x(x+4) + 4(x+4)(3x+13) = 256

7. Simplify the equation:
   16x^2 + 16sqrt(x+4)sqrt(3x+13) + 12x^2 + 48x + 16x + 48 + 12x^2 + 48x + 16x + 48 = 256
   40x^2 + 96x + 112 + 16sqrt(x+4)sqrt(3x+13) = 256

8. Move all terms to one side of the equation:
   40x^2 + 96x + 112 + 16sqrt(x+4)sqrt(3x+13) - 256 = 0

9. Simplify the equation further:
   40x^2 + 96x - 144 + 16sqrt(x+4)sqrt(3x+13) = 0

10. At this point, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula.