Question

Решите уравнение: (sqrt(2x+1)+1)/x=1.

Answer 1

ОДЗ х≠0; 2x+1≥0; x≥-1/2
умножим обе части на х
√(2х+1) +1=х
√(2х+1)=х-1
возведем обе части в квадрат
2х+1=х²-2х+1
x²-4x=0
x(x-4)=0
1) x=0 не подходит к одз
2) х=4
Ответ х=4
проверка
(√(2*4+1)+1)/4=(√9+1)/4=(3+1)/4=4/4=1

Answer 2

To solve the equation, let's follow these steps:

1. Multiply both sides of the equation by x to eliminate the denominator.
2. Simplify the equation.
3. Isolate the square root term on one side of the equation.
4. Square both sides to eliminate the square root.
5. Solve for x.
6. Check for extraneous solutions.

Let's start:

1. Multiply both sides by x: x * (sqrt(2x + 1) + 1) = x

2. Simplify the equation: sqrt(2x + 1) + x = x

3. Isolate the square root term: sqrt(2x + 1) = x - x

Since x - x = 0, the equation becomes:

sqrt(2x + 1) = 0

4. Square both sides: (2x + 1) = 0

5. Solve for x: 2x = -1 x = -1/2

Now, let's check for extraneous solutions by substituting the found value of x into the original equation:

(sqrt(2 * (-1/2) + 1) + 1) / (-1/2) = 1

(sqrt(-1 + 1) + 1) / (-1/2) = 1

(sqrt(0) + 1) / (-1/2) = 1

(0 + 1) / (-1/2) = 1

1 / (-1/2) = 1

This simplifies to:

-2 = 1

Since the equation is not true when x = -1/2, it means that the original equation has no real solutions. The solution x = -1/2 is an extraneous solution, which occurs when we squared both sides of the equation, and we must discard it. Therefore, the original equation has no real solutions.