SOLUTION: Find all real solutions to the equation {{{sqrt(x+4)-2sqrt(x-1)=-1)}}}

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Question 41521: Find all real solutions to the equation
sqrt%28x%2B4%29-2sqrt%28x-1%29=-1%29
Answer by Nate(3500) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28x%2B4%29-2sqrt%28x-1%29=-1%29
sqrt%28x%2B4%29+=+-1+%2B+2sqrt%28x-1%29
%28sqrt%28x%2B4%29%29%5E2+=+%28-1+%2B+2sqrt%28x-1%29%29%5E2
x+%2B+4+=+1+-+4sqrt%28x-1%29+%2B+4%28x-1%29
x+%2B+4+=+1+-+4sqrt%28x-1%29+%2B+4x+-+4
x+%2B+4+=+-4sqrt%28x-1%29+%2B+4x+-+3
-3x+%2B+7+=+-4sqrt%28x-1%29
%283%2F4%29x+-+%287%2F4%29+=+sqrt%28x-1%29
%28%283%2F4%29x+-+%287%2F4%29%29%5E2+=+%28sqrt%28x-1%29%29%5E2
%289%2F16%29x%5E2+-+%2842%2F16%29x+%2B+%2849%2F16%29+=+x+-+1
9x%5E2+-+42x+%2B+49+=+16x+-+16
9x%5E2+-+58x+%2B+65+=+0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 9x%5E2%2B-58x%2B65+=+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=%28-58%29%5E2-4%2A9%2A65=1024.

Discriminant d=1024 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--58%2B-sqrt%28+1024+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-58%29%2Bsqrt%28+1024+%29%29%2F2%5C9+=+5
x%5B2%5D+=+%28-%28-58%29-sqrt%28+1024+%29%29%2F2%5C9+=+1.44444444444444

Quadratic expression 9x%5E2%2B-58x%2B65 can be factored:
9x%5E2%2B-58x%2B65+=+9%28x-5%29%2A%28x-1.44444444444444%29
Again, the answer is: 5, 1.44444444444444. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-58%2Ax%2B65+%29

x+=+5 and x+=+13%2F9
Check:
sqrt%28x%2B4%29-2sqrt%28x-1%29=-1%29
sqrt%285%2B4%29-2sqrt%285-1%29=-1%29
sqrt%289%29-2sqrt%284%29=-1%29
3-2%282%29=-1%29
3-4=-1%29 Five Works
Check:
sqrt%28x%2B4%29-2sqrt%28x-1%29=-1%29
sqrt%28%2813%2F9%29%2B4%29-2sqrt%28%2813%2F9%29-1%29=-1%29
sqrt%28%2813%2F9%29%2B%2836%2F9%29%29-2sqrt%28%2813%2F9%29-%289%2F9%29%29=-1%29
sqrt%2849%2F9%29-2sqrt%284%2F9%29=-1%29
%287%2F3%29-2%282%2F3%29=-1%29
%287%2F3%29-%284%2F3%29=-1%29
%283%2F3%29=-1%29 (13/9) Does Not Work