SOLUTION: Hello, I've been trying to solve this equation but it seems I can only get down to simplifying it. Please help =) SOLVE THE EQUATION SYMBOLICALLY sqrt(x+6) + sqrt(2-x) = 4

Algebra ->  Radicals -> SOLUTION: Hello, I've been trying to solve this equation but it seems I can only get down to simplifying it. Please help =) SOLVE THE EQUATION SYMBOLICALLY sqrt(x+6) + sqrt(2-x) = 4      Log On


   

Question 85463: Hello, I've been trying to solve this equation but it seems I can only get down to simplifying it. Please help =)
SOLVE THE EQUATION SYMBOLICALLY
sqrt(x+6) + sqrt(2-x) = 4
Found 3 solutions by jim_thompson5910, scianci, Earlsdon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28x%2B6%29+%2B+sqrt%282-x%29+=+4 Start with the given equation

sqrt%28x%2B6%29=4+-+sqrt%282-x%29 Subtract sqrt%282-x%29 from both sides

x%2B6=%284+-+sqrt%282-x%29%29%5E2 Square both sides

x%2B6=16+-+8sqrt%282-x%29%2B2-x FOIL the right side

x%2B6-16-2%2Bx=+-+8sqrt%282-x%29Get everything but the term -+8sqrt%282-x%29 to the left side

2x-12=+-+8sqrt%282-x%29 Combine like terms on the left side

%282x-12%29%5E2=+64%2A%282-x%29 Square both sides

4x%5E2-48x%2B144=+64%2A%282-x%29 FOIL the left side

4x%5E2-48x%2B144=+128-64x Distribute

4x%5E2%2B16x%2B16=+0 Get everything to one side

4%28x%5E2%2B4x%2B4%29=+0 Factor out a 4

4%28x%2B2%29%28x%2B2%29=+0 Factor the expression in the parenthesis

So we have an answer of

x=-2

Check:
sqrt%28-2%2B6%29+%2B+sqrt%282--2%29+=+4 Plug in x=-2

sqrt%284%29+%2B+sqrt%284%29+=+4

2%2B2+=+4

4=4 solution works

Answer by scianci(186) About Me  (Show Source):
You can put this solution on YOUR website!
Isolate the radicals by subtracting one of them from both sides:
%28sqrt%28x%2B6%29%29 = 4 - %28sqrt%282-x%29%29
Square both sides:
%28sqrt%28x%2B6%29%29%5E2 = (4 - %28sqrt%282-x%29%29)^2
x + 6 = 16 - 8sqrt%282-x%29%29 + 2-x
x + 6 = 18 - 8sqrt%282-x%29%29-x
Isolate the remaining radical:
x + x + 6 - 18 = - 8sqrt%282-x%29%29
2x-12 = - 8sqrt%282-x%29%29
Square both sides again:
%282x-12%29%5E2 = (- 8%28sqrt%282-x%29%29)^2
4x%5E2 - 48x + 144 = 64(2 - x)
4x%5E2-48x+144 = 128-64x
4x%5E2+16x+16 = 0
x%5E2 + 4x + 4 = 0
(x + 2)(x + 2) = 0
x + 2 = 0
x = -2 [potentially]
Check for extraneous solutions:
%28sqrt%28-2%2B6%29%29 = 4 - %28sqrt%282-%28-2%29%29%29?
%28sqrt%284%29%29 = 4 - %28sqrt%284%29%29 ?
2 = 4 - 2 ?
2 = 2 ; OK
x = -2 is a valid solution

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Solve:
sqrt%28x%2B6%29+%2B+sqrt%282-x%29+=+4 First, separate the square roots. Subtract sqrt%282-x%29 from both sides.
sqrt%28x%2B6%29+=+4-sqrt%282-x%29 Next, square both sides.
x%2B6+=+16-8%2Asqrt%282-x%29%2B%282-x%29 Simplify so that you have only the radical sqrt%282-x%29 (or a multiple thereof) on one side.
x%2B6+=+18-8%2Asqrt%282-x%29-x Add x to both sides.
2x%2B6+=+18-8%2Asqrt%282-x%29 Subtract 18 from both sides.
2x-12+=+-8%2Asqrt%282-x%29 Now square both sides again.
4x%5E2+-+48x+%2B+144+=+64%282-x%29 Simplify this so that you have a quadratic equation in standard form.
4x%5E2+-+48x+%2B+144+=+128-64x Add 64x to both sides.
4x%5E2+%2B+16x+%2B144+=+128 Subtract 128 from both sides.
4x%5E2+%2B+16x+%2B+16+=+0 Factor out a 4 to simplify this.
4%28x%5E2+%2B+4x+%2B+4%29+=+0 Now factor the parentheses.
%28x%2B2%29%28x%2B2%29+=+0 Applying the zero products principle, you get the double root:
x+=+-2
Check:
sqrt%28x%2B6%29+%2B+sqrt%282-x%29+=+4 Set x = -2.
sqrt%28-2%2B6%29+%2B+sqrt%282-%28-2%29%29+=+4 Simplify.
sqrt%284%29+%2B+sqrt%284%29+=+4
2+%2B+2+=+4
4+=+4