Question

√(x+5/x) + 4√(x/x+5) = 4

Answer 1

Решение смотри на фото :)

Answer 2

Let's start by simplifying the equation as much as possible before solving for x.

First, we can simplify the square roots by writing them as exponents:

√(x+5/x) = (x+5/x)^(1/2)

4√(x/x+5) = 4(x/x+5)^(1/2)

Using a common denominator of x(x+5), we can combine the fractions:

(x+5)/x = (x^2 + 5x)/(x^2)

x/(x+5) = x/(x+5)

So the equation becomes:

(x+5/x)^(1/2) + 4(x/x+5)^(1/2) = 4

Subtracting 4 from both sides gives:

(x+5/x)^(1/2) + 4(x/x+5)^(1/2) - 4 = 0

Now, let's make a substitution to simplify the equation further. Let y = (x+5/x)^(1/2). Then:

4(x/x+5)^(1/2) = 4(1/y^2)^(1/2) = 4/y

Substituting back into the equation, we get:

y + 4/y = 4

Multiplying both sides by y, we get a quadratic equation:

y^2 - 4y + 4 = 0

Factoring, we get:

(y-2)^2 = 0

So y = 2. Substituting back to solve for x, we get:

(x+5/x)^(1/2) = 2

Squaring both sides, we get:

x+5/x = 4

Multiplying both sides by x, we get a quadratic equation:

x^2 - 4x + 5 = 0

Factoring, we get:

(x-1)(x-5) = 0

So x = 1 or x = 5.

Therefore, the solutions to the original equation are x = 1 and x = 5.