SOLUTION: square root of x-4 minus the square root of x-7 = 1

Algebra ->  Square-cubic-other-roots -> SOLUTION: square root of x-4 minus the square root of x-7 = 1      Log On


   

Question 121736: square root of x-4 minus the square root of x-7 = 1
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given:
.
sqrt%28x-4%29-sqrt%28x-7%29+=+1
.
To make it easier to solve for x, let's begin by taking one of the radicals to the other side.
We can get rid of the -sqrt%28x-7%29 on the left side by adding sqrt%28x-7%29 to both
sides. On the left side this addition cancels out the -sqrt%28x-7%29 and the equation
then becomes:
.
sqrt%28x-4%29+=+1+%2B+sqrt%28x-7%29
.
Now square both sides. If you do that the equation becomes:
.
x+-+4+=+1+%2B+2%2Asqrt%28x+-7%29+%2B+x+-+7
.
Let's next get rid of the terms that are just "x" on both sides by subtracting x from both
sides and the equation reduces to:
.
+-+4+=+1+%2B+2%2Asqrt%28x-7%29+-+7
.
On the right side combine the 1 and the -7 to get -6, making the equation:
.
+-+4+=+-6+%2B+2%2Asqrt%28x-7%29
.
Get rid of the -6 on the right side by adding +6 to both sides to further reduce the equation to:
.
2+=+2%2Asqrt%28x-7%29
.
Divide both sides of the equation by 2 to get rid of the "2s" and the equation is then:
.
1+=+sqrt%28x-7%29
.
Again, square both sides and the equation then is:
.
1+=+x+-+7
.
Get rid of the -7 on the right side by adding 7 to both sides and you finally have:
.
8+=+x
.
So the answer is x = 8. Let's check it, just to make sure. Return to the original given
equation:
.
sqrt%28x-4%29-sqrt%28x-7%29+=+1
.
Substitute 8 for x and the original equation becomes:
.
sqrt%288-4%29-sqrt%288-7%29+=+1
.
Simplify the numbers under the radicals and you have:
.
sqrt%284%29-sqrt%281%29+=+1
.
This simplifies to:
.
2+-+1+=+1
.
The left side equals the right side so the answer checks out OK. Our answer of x = 8 is correct.
.
Hope this helps you to see your way through the problem.
.