Question

Решить уравнение: (x + 5)⁴ - 10(x+5)²+9=0

Answer 1

Решала с помощью подстановки и теоремы Виета

Answer 2

To solve the equation (x + 5)⁴ - 10(x + 5)² + 9 = 0, we can use a substitution to simplify the equation. Let's set y = (x + 5)². Now the equation becomes:

y² - 10y + 9 = 0

This is a quadratic equation in y. To solve for y, we can use the quadratic formula:

y = (-b ± √(b² - 4ac)) / 2a

where a = 1, b = -10, and c = 9.

y = (10 ± √((-10)² - 4 * 1 * 9)) / 2 * 1 y = (10 ± √(100 - 36)) / 2 y = (10 ± √64) / 2 y = (10 ± 8) / 2

Now, we have two possible values for y:

1. y = (10 + 8) / 2 = 18 / 2 = 9
2. y = (10 - 8) / 2 = 2 / 2 = 1

Now, remember that y = (x + 5)². So, we have:

1. y = (x + 5)² = 9 Taking the square root of both sides: x + 5 = ±√9 x + 5 = ±3

1.a. x + 5 = 3 x = 3 - 5 x = -2

1.b. x + 5 = -3 x = -3 - 5 x = -8

2. y = (x + 5)² = 1 Taking the square root of both sides: x + 5 = ±√1 x + 5 = ±1

2.a. x + 5 = 1 x = 1 - 5 x = -4

2.b. x + 5 = -1 x = -1 - 5 x = -6

So, the solutions to the equation are x = -2, x = -8, x = -4, and x = -6.