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 x⁴ - 8x³ + 15x² - 2x - 9 = 0, we can use various methods. One of the common methods is to use numerical approaches or factorization. Let's try to factorize the equation first.

Step 1: Check for integer roots We can try to find integer roots by using the rational root theorem. According to the rational root theorem, any rational root of the equation must be of the form p/q, where p is a factor of the constant term (-9) and q is a factor of the leading coefficient (1).

The factors of -9 are ±1, ±3, and ±9. The factors of 1 are ±1.

Possible rational roots are: ±1, ±3, ±9.

Step 2: Test the possible roots Let's check these possible rational roots one by one using synthetic division or direct substitution to see if they satisfy the equation.

1. For x = 1: (1)⁴ - 8(1)³ + 15(1)² - 2(1) - 9 = 1 - 8 + 15 - 2 - 9 = -3 (not zero)

2. For x = -1: (-1)⁴ - 8(-1)³ + 15(-1)² - 2(-1) - 9 = 1 + 8 + 15 + 2 - 9 = 17 (not zero)

3. For x = 3: (3)⁴ - 8(3)³ + 15(3)² - 2(3) - 9 = 81 - 216 + 135 - 6 - 9 = -15 (not zero)

4. For x = -3: (-3)⁴ - 8(-3)³ + 15(-3)² - 2(-3) - 9 = 81 + 216 + 135 + 6 - 9 = 429 (not zero)

5. For x = 9: (9)⁴ - 8(9)³ + 15(9)² - 2(9) - 9 = 6561 - 5832 + 1215 - 18 - 9 = 947 (not zero)

6. For x = -9: (-9)⁴ - 8(-9)³ + 15(-9)² - 2(-9) - 9 = 6561 + 5832 + 1215 + 18 - 9 = 13617 (not zero)

None of the possible rational roots result in a zero value, so it seems that the equation does not have any rational roots. In such cases, we may resort to numerical methods like the Newton-Raphson method or graphical methods to find approximate solutions.

For the numerical approach, we can use numerical solvers available in software like Python or MATLAB. These solvers can approximate the roots of the equation. However, I cannot execute code directly here, so if you have access to a programming environment, you can use libraries like NumPy or SciPy in Python to find approximate solutions.

If you're interested, I can show you how to use Python to find approximate solutions using NumPy. Let me know if you would like to proceed with that approach.