Question

2(x+5)^1/2=7(x+5)^1/4-6

Answer 1

Используя метод замены ищем значение переменной а, а оттуда подставляем его в уравнение вида (х+5)^1/4=а и вычисляем значение х

Answer 2

To solve the equation 2(x+5)^(1/2) = 7(x+5)^(1/4) - 6, we can use a substitution to simplify the equation. Let's substitute a new variable, u = (x+5)^(1/4).

Using this substitution, we have: 2u^2 = 7u - 6

Rearranging the equation: 2u^2 - 7u + 6 = 0

Now, we can solve this quadratic equation. Factoring it, we have: (2u - 3)(u - 2) = 0

Setting each factor to zero and solving for u: 2u - 3 = 0 --> u = 3/2 u - 2 = 0 --> u = 2

Now, we substitute back u = (x+5)^(1/4): For u = 3/2: (x+5)^(1/4) = 3/2

Raising both sides of the equation to the power of 4: [(x+5)^(1/4)]^4 = (3/2)^4 x + 5 = (81/16) x = 81/16 - 80/16 x = 1/16

For u = 2: (x+5)^(1/4) = 2

Raising both sides of the equation to the power of 4: [(x+5)^(1/4)]^4 = 2^4 x + 5 = 16 x = 16 - 5 x = 11

Therefore, the solutions to the equation are x = 1/16 and x = 11.