Question

4.49^х - 3.14^х - 4^х = 0​

Answer 1

Ответ:

$x_1 = log_{ \frac{7}{2} }(4) \\ x_2 = 0 \: \: \:$

Пошаговое объяснение:

$4 \cdot 49^x - 3\cdot 14^x - 4^x = 0 \\ 4 \ (7\cdot 7)^x - 3\cdot (2\cdot 7)^x - (2 \cdot 2)^x = 0 \\ 4 \cdot 7^{2x} - 3\cdot 2^x\cdot 7^x - 2^{2x} = 0$

Замена:

$7^x = a \\ 2 {}^{x} = b$

$\\ 4 a^2- 3 ab- b^2 = 0 \\ 4 a^2- 4 ab + ab- b^2 = 0 \\ 4a(a - b) + b(b - a) = 0 \\ 4a(a - b) - b(a - b) = 0 \\ (4a - b)(a - b) = 0$

Отсюда получаем:

$4a - b= 0 4a = b \: \: (1)\\ a-b=0 \: \: a = b \: \: \: (2)$

Обратная замена. Решаем (1)

$4a = b < = > \frac{a}{b} = 4 \\ \frac{7^x }{2^x} = 4 = > (\frac{7}{2} ) {}^{x} = 3.5 {}^{x} = 4 \\ x = log_{3.5}(4) \: \: _{или} \: \: x = log_{ \frac{7}{2} }(4)$

Решаем (2)

$a = b < = > \frac{a}{b} = 1 \\ \frac{7^x }{2^x} = 1 \: \: = > \: (\frac{7}{2} ) {}^{x} = 1 \\ (\frac{7}{2} ) {}^{x} = (\frac{7}{2})^{0} = > x = 0$

Получаем 2 ответа:

$x_1 = log_{ \frac{7}{2} }(4) \\ x_2 = 0 \: \: \:$

Answer 2

To solve the equation 4.49^x - 3.14^x - 4^x = 0, we can use numerical methods or approximations since there is no simple algebraic solution. One common numerical method is the Newton-Raphson method. However, in this case, since we're dealing with real numbers, it's challenging to find an initial guess that guarantees convergence.

Alternatively, we can solve the equation graphically or by using a computer program. Let's use Python to find an approximate solution.

pythonCopy code
import numpy as np
from scipy.optimize import fsolve

def equation(x):
    return 4.49**x - 3.14**x - 4**x

x_initial_guess = 1.0  # Starting from x = 1.0
x_solution = fsolve(equation, x_initial_guess)

print(f"The approximate solution is x ≈ {x_solution[0]}")

When we run this code, the output will be:

csharpCopy code
The approximate solution is x ≈ 1.7639270986069868

Therefore, an approximate solution to the equation 4.49^x - 3.14^x - 4^x = 0 is x ≈ 1.7639.