How do you solve #sqrt(2x+1)+sqrt(2x+6)=5#?

Question

Abigail Allen · Accepted Answer

x= 3/2

Try move one square to the other side: #sqrt(2x+1)=5-sqrt(2x+6)#
Take a square and you will get: #2x+1=25+2x+6-10*sqrt(2x+6)#
And now the 2xs are cancelled on both sides, rearrange and you will get: #10*sqrt(2x+6)=30#
Now the equation is simple, take a square on both sides and you will get the answer.

Savannah Anderson · Answer

undefined To solve the equation sqrt(2x+1) + sqrt(2x+6) = 5, you can follow these steps:

1. Start by isolating one of the square root terms. Subtract sqrt(2x+6) from both sides of the equation:
   sqrt(2x+1) = 5 - sqrt(2x+6)

2. Square both sides of the equation to eliminate the square root on the left side:
   (sqrt(2x+1))^2 = (5 - sqrt(2x+6))^2
   2x + 1 = 25 - 10sqrt(2x+6) + 2x + 6

3. Simplify the equation by combining like terms:
   2x + 1 = 31 - 10sqrt(2x+6)

4. Move all terms involving the square root to one side of the equation:
   10sqrt(2x+6) = 30 - 2x

5. Square both sides of the equation again to eliminate the square root:
   (10sqrt(2x+6))^2 = (30 - 2x)^2
   100(2x+6) = 900 - 120x + 4x^2

6. Expand and simplify the equation:
   200x + 600 = 900 - 120x + 4x^2

7. Rearrange the equation to form a quadratic equation:
   4x^2 + 320x - 300 = 0

8. Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, the quadratic equation can be factored as:
   (2x - 5)(2x + 60) = 0

9. Set each factor equal to zero and solve for x:
   2x - 5 = 0  or  2x + 60 = 0

10. Solve for x in each equation:
   x = 5/2  or  x = -30

Therefore, the solutions to the equation sqrt(2x+1) + sqrt(2x+6) = 5 are x = 5/2 and x = -30.