SOLUTION: Can you please solve this equation: sqrt( x + 5) = sqrt(x^2 - 15)

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Question 246169: Can you please solve this equation:
sqrt( x + 5) = sqrt(x^2 - 15)
Found 2 solutions by unlockmath, richwmiller:
Answer by unlockmath(1688) About Me  (Show Source):
You can put this solution on YOUR website!
Hello,
If we square each side then the square roots cancel.
[sqrt( x + 5)]^2 = [sqrt(x^2 - 15)]^2 reults in:
x + 5 = x^2 - 15 Now move the x + 5 to the other side:
0=x^2-x-20 Now we can factor this to be:
0= (x-5)(x+4)This gives us:
x=5
x=-4
These can be put into the original problem.
Does it work out?
RJ Toftness
Check out my book at www.math-unlock.com

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt( x + 5) = sqrt(x^2 - 15)
square both sides
x+5=x^2-15
subtract x+5
0=x^2-x-10
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-1x%2B-10+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-1%29%5E2-4%2A1%2A-10=41.

Discriminant d=41 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1%2B-sqrt%28+41+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1%29%2Bsqrt%28+41+%29%29%2F2%5C1+=+3.70156211871642
x%5B2%5D+=+%28-%28-1%29-sqrt%28+41+%29%29%2F2%5C1+=+-2.70156211871642

Quadratic expression 1x%5E2%2B-1x%2B-10 can be factored:
1x%5E2%2B-1x%2B-10+=+1%28x-3.70156211871642%29%2A%28x--2.70156211871642%29
Again, the answer is: 3.70156211871642, -2.70156211871642. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-1%2Ax%2B-10+%29

we can also solve by completing the square
add 10 to both sides
10=x^2-x
add (1/2)^2 to both sides
41/4=x^2-x+1/4
factor
41/4=(x-1/2)^2
get sqrt
sqrt(41)/2=x-1/2
-sqrt(41)/2=x-1/2
add 1/2 to both sides
1/2-sqrt(41)/2=x
1/2+sqrt(41)/2=x
factor
1/2(1-sqrt(41))=x
1/2(1+sqrt(41))=x