Question

Решите уравнение: (x-2)⁴ - (x-2)² -12 =0

Answer 1

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Answer 2

$d = 49 \\ u1 = 4 \\ u2 = - 3$
$(x - 2) {}^{2} = - 3 \\ x3 = 2 + \sqrt{3} \\ x4 = 2 - \sqrt{3}$

$(x - 2) {}^{2} = 2 \\ x1 = 0 \\ x2 = 4$

Answer 3

To solve the equation (x-2)⁴ - (x-2)² - 12 = 0, we can use a substitution to simplify the equation.

Let's make the substitution: y = (x-2)²

Now the equation becomes a quadratic equation in terms of y:

y² - y - 12 = 0

Now, we can factor the quadratic equation:

(y - 4)(y + 3) = 0

Now, set each factor to zero and solve for y:

1. y - 4 = 0 y = 4

2. y + 3 = 0 y = -3

Now, remember that y = (x-2)². So, we have:

1. (x-2)² = 4

To find the solutions for x, take the square root of both sides:

x - 2 = ±√4

1. x - 2 = 2 x = 2 + 2 x = 4

2. x - 2 = -2 x = 2 - 2 x = 0

Now, let's solve the second equation:

2. (x-2)² = -3

Since the square of any real number is always non-negative, there are no real solutions for this part of the equation.

Therefore, the solutions to the original equation are x = 4 and x = 0.