Question

√2x+1-√x-1=1 розв'язати вправи​

Answer 1

To solve the equation √(2x+1) - √(x-1) = 1, we can start by isolating one of the square roots. Let's isolate √(2x+1):

√(2x+1) = 1 + √(x-1)

Now, square both sides of the equation to eliminate the square root:

(√(2x+1))^2 = (1 + √(x-1))^2

Simplifying,

2x + 1 = 1 + 2√(x-1) + (x-1)

Combine like terms,

2x + 1 = 2√(x-1) + x

Now, let's isolate the square root term by moving all other terms to the left side:

2√(x-1) - x = -2x + 1

Next, square both sides again to eliminate the square root:

(2√(x-1) - x)^2 = (-2x + 1)^2

Expanding and simplifying,

4(x-1) - 4x√(x-1) + x^2 = 4x^2 - 4x + 1

Rearrange the equation to one side:

0 = 4x^2 - 5x - 3 + 4x√(x-1) - 4(x-1)

Combine like terms,

0 = 4x^2 - x - 7 + 4x√(x-1)

Now, let's isolate the square root term:

4x√(x-1) = x + 7 - 4x^2

Square both sides once more:

(4x√(x-1))^2 = (x + 7 - 4x^2)^2

Expanding and simplifying,

16x^2(x-1) = x^2 + 49 + 16x^4 - 14x - 28x^3

Rearrange the equation to one side:

0 = 16x^4 - 28x^3 + 15x^2 - 15x + 49

Unfortunately, this equation cannot be solved algebraically. You would need to use numerical methods or graphing to find an approximate solution.