SOLUTION: Please help me to solve this equation the values of x satisfying the equation √(2x²+5x-2) - √(2x²+5x-9) = 1 are

Algebra ->  Radicals -> SOLUTION: Please help me to solve this equation the values of x satisfying the equation √(2x²+5x-2) - √(2x²+5x-9) = 1 are      Log On


   

Question 895168: Please help me to solve this equation
the values of x satisfying the equation
√(2x²+5x-2) - √(2x²+5x-9) = 1 are
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

sqrt%282x%5E2%2B5x-2%29+-+sqrt%282x%5E2%2B5x-9%29+=+1

Isolate one of the terms with a radical:

sqrt%282x%5E2%2B5x-2%29+=+1%2B+sqrt%282x%5E2%2B5x-9%29+

Square both sides:

%28sqrt%282x%5E2%2B5x-2%29%29%5E2+=+%281%2B+sqrt%282x%5E2%2B5x-9%29%29%5E2+

To make things a little easier to see

let A=sqrt%282x%5E2%2B5x-2%29 and let  B=+sqrt%282x%5E2%2B5x-9%29+

Then we have

A%5E2+=+%281%2B+B%29%5E2+

A%5E2+=+%281%2BB%29%281%2BB%29

A%5E2+=+1%2B2B%2BB%5E2

Since we let A=sqrt%282x%5E2%2B5x-2%29 and let B=+sqrt%282x%5E2%2B5x-9%29+

then A%5E2=2x%5E2%2B5x-2 and  B%5E2=2x%5E2%2B5x-9+

So

A%5E2+=+1%2B2B%2BB%5E2

becomes

2x%5E2%2B5x-2=1%2B2sqrt%282x%5E2%2B5x-9%29+%2B+2x%5E2%2B5x-9

2x%5E2%2B5x-2=-8%2B2sqrt%282x%5E2%2B5x-9%29+%2B+2x%5E2%2B5x

Isolate the radical term (the 2x²'s and the 5x's will cancel)

-2=-8%2B2sqrt%282x%5E2%2B5x-9%29

6=2sqrt%282x%5E2%2B5x-9%29

Divide both sides by 2

3=sqrt%282x%5E2%2B5x-9%29

Square both sides:

9=2x%5E2%2B5x-9

Get 0 on the left:

0=2x%5E2%2B5x-18

Factor the right side:

0=%282x%2B9%29%28x-2%29

Use the zero-factor property:

2x+9=0;     x-2=0
  2x=-9       x=2
   x=-9%2F2

But we must check in the original equation, since radical 
equations often have extraneous answers:

Checking x=-9%2F2

sqrt%282x%5E2%2B5x-2%29+-+sqrt%282x%5E2%2B5x-9%29+=+1



sqrt%282%2881%2F4%29-45%2F2-2%29+-+sqrt%282%2881%2F4%29-45%2F2-9%29+=+1

sqrt%28162%2F4-45%2F2-2%29+-+sqrt%28162%2F4-45%2F2-9%29+=+1

sqrt%28162%2F4-90%2F4-8%2F4%29+-+sqrt%28162%2F4-90%2F4-36%2F4%29+=+1

sqrt%2864%2F4%29+-+sqrt%2836%2F4%29+=+1

sqrt%2816%29+-+sqrt%289%29+=+1

4-3=1

1=1

So -9%2F2 is a solution.

Checking x=2

sqrt%282x%5E2%2B5x-2%29+-+sqrt%282x%5E2%2B5x-9%29+=+1

sqrt%282%282%29%5E2%2B5%282%29-2%29+-+sqrt%282%282%29%5E2%2B5%282%29-9%29+=+1

sqrt%282%284%29%2B10-2%29+-+sqrt%282%284%29%2B10-9%29+=+1
sqrt%288%2B8%29+-+sqrt%288%2B1%29+=+1

sqrt%2816%29+-+sqrt%289%29+=+1

4-3=1

1=1

So x=2 is also a solution.

Edwin