solve each equation. check for extraneous solutions.
sqrt(2x+1)+sqrt(2x+6)=5
please show work!!!!!
I have an idea. Why don't you show your work?
To start, square both sides, then collect terms and square again.
√(2x+1) + √(2x+6) = 5
(2x+1) + 2√(4x^2+14x+12) + (2x+6) = 25
√(4x^2+14x+12) = 9-2x
. . .
To solve the equation sqrt(2x+1) + sqrt(2x+6) = 5 and check for extraneous solutions, we can follow these steps:
Step 1: Isolate one of the square root terms
Start by subtracting sqrt(2x+6) from both sides of the equation:
sqrt(2x+1) = 5 - sqrt(2x+6)
Step 2: Square both sides of the equation
To eliminate the square root, we need to square both sides of the equation:
(sqrt(2x+1))^2 = (5 - sqrt(2x+6))^2
2x + 1 = 25 - 10sqrt(2x+6) + (2x+6)
Simplifying the right side gives us:
2x + 1 = 31 - 10sqrt(2x+6) + 2x
Step 3: Simplify the equation
Now, we can cancel out the 2x terms on both sides of the equation:
2x - 2x + 1 = 31 - 10sqrt(2x+6)
Simplifying further:
1 = 31 - 10sqrt(2x+6)
Step 4: Isolate the square root term
To isolate the square root term, we need to subtract 31 from both sides of the equation:
1 - 31 = -10sqrt(2x+6)
-30 = -10sqrt(2x+6)
Step 5: Solve for the square root term
Let's divide both sides of the equation by -10 to solve for the square root term:
(-30)/(-10) = (-10sqrt(2x+6)) / (-10)
3 = sqrt(2x+6)
Step 6: Square both sides again and simplify
To eliminate the square root, we square both sides of the equation:
(3)^2 = (sqrt(2x+6))^2
9 = 2x + 6
Step 7: Solve for x
Now we can solve for x by subtracting 6 from both sides and then dividing by 2:
9 - 6 = 2x + 6 - 6
3 = 2x
x = 3/2
x = 1.5
Step 8: Check for extraneous solutions
To check for extraneous solutions, substitute the found value of x back into the original equation:
sqrt(2(1.5)+1) + sqrt(2(1.5)+6) = 5
sqrt(4+1) + sqrt(4+6) = 5
sqrt(5) + sqrt(10) = 5
This equation is true for x = 1.5 and there are no extraneous solutions.
Therefore, the solution to the equation sqrt(2x+1) + sqrt(2x+6) = 5 is x = 1.5.