Solve:
squareroot 2x-1 - squareroot x+3 = 1
show steps
To solve the given equation √(2x-1) - √(x+3) = 1, we can follow these steps:
Step 1: Isolate one square root term on one side of the equation.
Start by moving the √(x+3) term to the right side of the equation:
√(2x - 1) = √(x+3) + 1
Step 2: Square both sides of the equation to get rid of the square root.
Squaring both sides will give us:
(√(2x-1))^2 = (√(x+3) + 1)^2
2x - 1 = (√(x+3) + 1) × (√(x+3) + 1)
2x - 1 = (x + 3) + 2√(x + 3) + 1
Step 3: Simplify the right side.
Combine like terms on the right side:
2x - 1 = x + 4 + 2√(x + 3)
Step 4: Isolate the square root term.
Move the x term to the left side of the equation and the constant terms to the right side:
2x - x - 1 - 4 = 2√(x + 3)
x - 5 = 2√(x + 3)
Step 5: Square both sides again.
We need to square both sides of the equation to eliminate the square root:
(x - 5)^2 = (2√(x + 3))^2
(x - 5)^2 = 4(x + 3)
Step 6: Expand and simplify.
Distribute the square on the left side:
x^2 - 10x + 25 = 4x + 12
Step 7: Move all terms to one side of the equation.
Subtract 4x and 12 from both sides:
x^2 - 10x - 4x + 25 - 12 = 0
x^2 - 14x + 13 = 0
Step 8: Factor or use the quadratic formula to solve for x.
We can factor the quadratic expression:
(x - 13)(x - 1) = 0
Step 9: Solve for x.
Set each factor to zero and solve for x:
x - 13 = 0 or x - 1 = 0
x = 13 or x = 1
Therefore, the solutions to the equation are x = 13 and x = 1.