Question

Как решить рациональное уравнение ?x^{4}-3 x^{2}-4=0

Answer 1

Пусть x^2 = y
x^4 = y^2

y^2 -3y -4 = 0

D = (-3)*2 - 4·1·(-4) = 9 + 16 = 25

y1 = (3 - √25)/2·1 = (3 - 5)/2 = -2/2 = -1

y2 = (3 + √25)/2·1 = (3 + 5)/2 = 8/2 = 4

x^2 = y

x^2 = -1(не имеет решения) 

x^2 = 4 

 х= 2  или х =  -2

Answer 2

Solving the Rational Equation

To solve the rational equation x^4 - 3 + x^2 - 4 = 0, we can follow these steps:

1. Combine like terms to simplify the equation: x^4 + x^2 - 7 = 0.

2. Factor the equation if possible. However, in this case, factoring is not straightforward, so we'll use an alternative method.

3. Substitute a variable to simplify the equation. Let's use y = x^2. By substituting, the equation becomes y^2 + y - 7 = 0.

4. Solve the quadratic equation y^2 + y - 7 = 0. We can use the quadratic formula or factorization to solve it. However, in this case, the quadratic formula is the most suitable method.

The quadratic formula is given by y = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.

Applying the quadratic formula to our equation, we have:

y = (-1 ± √(1^2 - 4(1)(-7))) / (2(1))

Simplifying further:

y = (-1 ± √(1 + 28)) / 2

y = (-1 ± √29) / 2

Therefore, the two possible values for y are (-1 + √29) / 2 and (-1 - √29) / 2.

5. Substitute back the value of y into the equation y = x^2 to find the values of x.

For y = (-1 + √29) / 2:

(-1 + √29) / 2 = x^2

Taking the square root of both sides:

x = ±√((-1 + √29) / 2)

For y = (-1 - √29) / 2:

(-1 - √29) / 2 = x^2

Taking the square root of both sides:

x = ±√((-1 - √29) / 2)

Therefore, the solutions to the rational equation x^4 - 3 + x^2 - 4 = 0 are:

x = ±√((-1 + √29) / 2) and x = ±√((-1 - √29) / 2).

Please note that these solutions are approximate values and can be further simplified.