Question

√(x-2)-√(2x+2)=√(2x-5)-√(3x-1)

Answer 1

Ответ: х=3. Решение в приложении, должно быть понятно

Answer 2

To solve the equation √(x-2) - √(2x+2) = √(2x-5) - √(3x-1), we can follow these steps:

Step 1: Get rid of square roots from the denominators by squaring both sides of the equation. However, keep in mind that squaring an equation may introduce extraneous solutions, so we must be careful to check our solutions later.

Step 2: Simplify the equation and collect all terms on one side.

Step 3: Factorize the equation and solve for x.

Step 4: Verify that the solutions obtained do not cause any denominators to be zero (since that would result in undefined values).

Let's proceed with the steps:

Step 1: Square both sides of the equation:

(√(x-2) - √(2x+2))^2 = (√(2x-5) - √(3x-1))^2

Step 2: Expand the squared terms:

(x - 2) - 2√((x-2)(2x+2)) + (2x + 2) = (2x - 5) - 2√((2x-5)(3x-1)) + (3x - 1)

Step 3: Move all terms to one side of the equation:

x - 2x - 2x + 2 - 2√((x-2)(2x+2)) + 2√((2x-5)(3x-1)) + 5x - 1 = 0

Combine like terms:

6x - 2 - 2√((x-2)(2x+2)) + 2√((2x-5)(3x-1)) = 0

Step 4: Now, isolate the square root terms on one side:

2√((x-2)(2x+2)) - 2√((2x-5)(3x-1)) = 6x - 2

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

(2√((x-2)(2x+2)) - 2√((2x-5)(3x-1)))^2 = (6x - 2)^2

Expand:

4((x-2)(2x+2)) - 4√((x-2)(2x+2))√((2x-5)(3x-1)) + 4((2x-5)(3x-1)) = 36x^2 - 24x + 4

Now, bring all terms to one side of the equation:

4((x-2)(2x+2)) - 4√((x-2)(2x+2))√((2x-5)(3x-1)) + 4((2x-5)(3x-1)) - 36x^2 + 24x - 4 = 0

Divide the entire equation by 4 to simplify:

((x-2)(2x+2)) - √((x-2)(2x+2))√((2x-5)(3x-1)) + ((2x-5)(3x-1)) - 9x^2 + 6x - 1 = 0

Now, let's define a variable substitution for clarity:

Let A = (x-2)(2x+2) and B = (2x-5)(3x-1).

The equation now becomes:

A - √(AB) + B - 9x^2 + 6x - 1 = 0

Now we have a quadratic equation in terms of A and B. Let's solve for A and B:

√(AB) = A + B - 9x^2 + 6x - 1

Square both sides again:

AB = (A + B - 9x^2 + 6x - 1)^2

Expand:

AB = A^2 + B^2 + 81x^4 - 36x^3 + 12x^2 - 2AB - 2A^2 + 2A(6x - 1) - 2B(6x - 1)

Combine like terms:

AB + 2AB = A^2 + B^2 - 2A^2 + 81x^4 - 36x^3 + 12x^2 + 2A(6x - 1) - 2B(6x - 1)

3AB = B^2 - A^2 + 81x^4 - 36x^3 + 12x^2 + 2A(6x - 1) - 2B(6x - 1)

Now, substitute the expressions for A and B back into the equation:

3(x-2)(2x+2)(2x-5)(3x-1) = ((2x-5)(3x-1))^2 - ((x-2)(2x+2))^2 + 81x^4 - 36x^3 + 12x^2 + 2(x-2)(2x+2)(6x-1) - 2(2x-5)(3x-1)(6x-1)

Now we have a cubic equation in terms of x. Solving this cubic equation will give us the values of x that satisfy the original equation.

However, solving this cubic equation directly can be quite complex and may not lead to an easy, closed-form solution. In many cases, numerical methods or approximation techniques are used to find the roots of such equations.

Keep in mind that due to the squaring operations, some solutions might be extraneous and not satisfy the original equation. Therefore, it's important to check any obtained solutions back into the original equation to verify their validity.