Question

1/(x-2)^2+9/(x+2)^2-6/x^2-4=0​

Answer 1

Ответ:

х=4

Пошаговое объяснение:

Умножим все на x^2-4=(x-2)(x+2) и

обозначим (x+2)/(x-2)=у

получим

у+9(1/у)=6

умножим на у

y^2-6y+9=0

(y-3)^2=0

y=3

(x+2)/(x-2)=3

x+2=3x-6

2x=8

x=4    входит в ОДЗ

Ответ: х=4

Answer 2

To solve the equation:

1/((x-2)^2) + 9/((x+2)^2) - 6/(x^2 - 4) = 0

First, let's simplify the equation by finding a common denominator for the fractions:

1/((x-2)^2) + 9/((x+2)^2) - 6/((x-2)(x+2)(x+2)(x-2)) = 0

Now, let's find a common denominator for the fractions on the left side:

[(x+2)^2(x-2)^2 + 9(x-2)^2 - 6]/((x-2)^2(x+2)^2) = 0

Now, we have a single fraction on the left side:

[(x+2)^2(x-2)^2 + 9(x-2)^2 - 6] = 0

Next, expand and simplify the equation:

(x^2 - 4)(x^2 - 4) + 9(x^2 - 4) - 6 = 0

Now, let's simplify further:

(x^4 - 8x^2 + 16) + 9x^2 - 36 - 6 = 0

Combine like terms:

x^4 + x^2 - 26 = 0

Now, this is a quartic equation. Let's make a substitution to simplify it. Let y = x^2:

y^2 + y - 26 = 0

Now, we can solve for y using the quadratic formula:

y = (-1 ± √(1^2 - 41(-26))) / (2*1)

y = (-1 ± √(1 + 104)) / 2

y = (-1 ± √105) / 2

Now, we have two possible values for y:

1. y = (-1 + √105) / 2
2. y = (-1 - √105) / 2

Now, we need to solve for x:

1. For y = (-1 + √105) / 2: x^2 = (-1 + √105) / 2 x = ±√[(-1 + √105) / 2]

2. For y = (-1 - √105) / 2: x^2 = (-1 - √105) / 2 x = ±√[(-1 - √105) / 2]

So, there are four possible solutions for x, taking into account both the positive and negative square roots of the values of y:

1. x = √[(-1 + √105) / 2]
2. x = -√[(-1 + √105) / 2]
3. x = √[(-1 - √105) / 2]
4. x = -√[(-1 - √105) / 2]

These are the solutions to the original equation.