Question

Решить уравнениеx⁴ - 8x³ + 15x² - 2x - 9 = 0

Answer 1

X⁴ - 8x³ + 15x² - 2x - 9 = 0
x⁴ - 8x³ + 24x² - 32x + 16 - 9x² + 30x - 25 = 0
(x⁴ - 4*x³*2 + 6*x²*2² - 4*x*2³ + 2⁴) - ( (3*x)² - 2*3x*5 + 5²) = 0
(x - 2)⁴ - (3x - 5)² = 0
( (x - 2)² - (3x - 5) )( (x - 2)² + (3x - 5) ) = 0
(x² - 4x + 4 - 3x + 5)(x² - 4x + 4 + 3x - 5) = 0
(x² - 7x + 9)(x² - x - 1) = 0

1) x² - 7x + 9 = 0
D = (-7)² - 4 * 9 = 49 - 36 = 13
$x_{1,2}=\frac{7\pm \sqrt{13}}{2}$

2) x² - x - 1 = 0
D = (-1)² - 4 * (-1) = 1 + 4 = 5
$x_{3,4}=\frac{1\pm \sqrt{5}}{2}$

Ответ: $\frac{7\pm \sqrt{13}}{2}; \frac{1\pm \sqrt{5}}{2}$

Answer 2

To solve the equation $x^4 - 8x^3 + 15x^2 - 2x - 9 = 0$, we can use various methods such as factoring, synthetic division, or numerical methods. However, this particular equation does not have any rational roots. Therefore, the most straightforward method to find approximate solutions is by using numerical methods like the Newton-Raphson method.

Before we proceed, let's define the function for which we want to find the roots:

$f(x) = x^4 - 8x^3 + 15x^2 - 2x - 9$.

Now, let's use the Newton-Raphson method to approximate the solutions. The Newton-Raphson method is an iterative numerical technique to find the roots of a real-valued function. The formula for this method is:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$,

where $x_{n+1}$ is the next approximation, $x_n$ is the current approximation, $f(x_n)$ is the value of the function at $x_n$, and $f'(x_n)$ is the derivative of the function at $x_n$.

To use this method, we need to start with an initial approximation of the root. Let's choose $x_0 = 2$ as our initial guess.

Step 1: Find the derivative of the function $f(x)$:

$f'(x) = 4x^3 - 24x^2 + 30x - 2$.

Step 2: Apply the Newton-Raphson formula to find the next approximation $x_1$:

$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$

$x_1 = 2 - \frac{f(2)}{f'(2)}$

Step 3: Repeat the process until the desired level of accuracy is achieved.

Now, let's go through the iterations:

Iteration 1: $x_0 = 2$, $x_1 = 2 - \frac{f(2)}{f'(2)}$.

We substitute the values of $x_0$ and $x_1$ into the equation:

$x_1 = 2 - \frac{(2)^4 - 8(2)^3 + 15(2)^2 - 2(2) - 9}{4(2)^3 - 24(2)^2 + 30(2) - 2}$.

Calculate $x_1$ to get the next approximation.

Repeat the process until you get a satisfactory approximation for $x$. You can use a calculator or a computer program to perform the iterations.

Note: The Newton-Raphson method may converge to different roots depending on the initial guess. If there are multiple roots, you may need to try different initial approximations to find all of them. Additionally, if the function has complex roots, you would need to use a more advanced numerical method.