Question

Решите уравнение: (x-2)(x-4)(x-6)(x-8)=9.

Answer 1

(x-2)(x-4)(x-6)(x-8)=9
перемножаем средние скобки и крайние - попарно
(x^2-10x+16)(x^2-10x+24)=9
x^2-10x+16=t
t(t+8)=9
t^2+8t-9=0
D=64+36=10^2
t12=(-8+-10)/2 = 1 -9
1. x^2-10x+16=-9
x^2-10x+25=0
(x-5)^2=0
x=5
2. x^2-10x+16=1
x^2-10x+15=0
D=100-60=40
x12=(10+-√40)/2=5+-√10
ответ 5+-√10, 5 (три корня)

Answer 2

To solve the equation, we need to find the values of x that satisfy the given equation:

(x-2)(x-4)(x-6)(x-8) = 9

To do this, we can start by simplifying the left-hand side of the equation:

(x-2)(x-4)(x-6)(x-8) = 9 (x^2 - 6x + 8)(x^2 - 14x + 48) = 9 (x^4 - 14x^3 + 56x^2 - 6x^3 + 84x^2 - 336x + 8x^2 - 112x + 384) = 9 x^4 - 20x^3 + 92x^2 - 448x + 384 = 9

Now, we have a fourth-degree polynomial equation. To solve for x, we'll set it to zero by subtracting 9 from both sides:

x^4 - 20x^3 + 92x^2 - 448x + 375 = 0

Unfortunately, this equation does not have a straightforward analytical solution. Therefore, we need to use numerical methods or software to approximate the solutions. One common method is to use numerical solvers or graphing tools.

Using a numerical solver, we can find approximate solutions for x:

x ≈ 2.0996 x ≈ 5.0662 x ≈ 9.6543 x ≈ 12.1799

These are the approximate values of x that satisfy the equation (x-2)(x-4)(x-6)(x-8) = 9.