2x-x в кубе=-8
Let's solve the equation:
\[ 2x - x^3 = -8 \]
Step 1: Rewrite the equation
\[ -x^3 + 2x = -8 \]
Step 2: Rearrange to standard form
Bring all terms to one side: \[ -x^3 + 2x + 8 = 0 \]
Multiply through by -1 to make it more straightforward: \[ x^3 - 2x - 8 = 0 \]
Step 3: Find possible rational roots
Using the Rational Root Theorem, possible rational roots are factors of 8 over factors of 1:
\[ \pm 1, \pm 2, \pm 4, \pm 8 \]
Step 4: Test possible roots
Test \( x = 2 \):
\[ (2)^3 - 2(2) - 8 = 8 - 4 - 8 = -4 \neq 0 \]
Test \( x = 4 \):
\[ (4)^3 - 2(4) - 8 = 64 - 8 - 8 = 48 \neq 0 \]
Test \( x = -2 \):
\[ (-2)^3 - 2(-2) - 8 = -8 + 4 - 8 = -12 \neq 0 \]
Test \( x = -1 \):
\[ (-1)^3 - 2(-1) - 8 = -1 + 2 - 8 = -7 \neq 0 \]
Test \( x = 8 \):
\[ 8^3 - 2(8) - 8 = 512 - 16 - 8 = 488 \neq 0 \]
Test \( x = -4 \):
\[ (-4)^3 - 2(-4) - 8 = -64 + 8 - 8 = -64 \neq 0 \]
Step 5: Use synthetic division or factorization
Since rational roots did not work, try factoring or numerical methods.
Step 6: Numerical approach or factoring technique
Let's test \( x = 2 \) again carefully: \[ 2^3 - 2(2) - 8 = 8 - 4 - 8 = -4 \neq 0. \]
Try \( x = 3 \):
\[ 3^3 - 2(3) - 8 = 27 - 6 - 8 = 13 \neq 0. \]
Try \( x = 1 \):
\[ 1 - 2 - 8 = -9 \neq 0. \]
Try \( x = -3 \):
\[ -27 + 6 - 8 = -29 \neq 0. \]
Step 7: Approximate numerical solutions
Since rational roots are not evident, use approximate methods (like the quadratic formula for depressed quadratic or graphing). Alternatively, note that:
- When \( x = 2 \):
\[ x^3 - 2x - 8 \approx -4 \]
- When \( x = 3 \):
\[ \approx 13 \]
Since the function crosses zero between 2 and 3, there is a root around \( x \approx 2.2 \).
Summary:
The exact roots are not rational; approximate solutions can be found numerically:
\[ x \approx 2.2 \]
Alternatively, solve using a cubic solver or calculator for more precise solutions.
Final note: The equation \( 2x - x^3 = -8 \) simplifies to:
\[ x^3 - 2x - 8 = 0 \]
which can be solved numerically or graphically to find approximate roots near \( x \approx 2.2 \).
Let me know if you'd like a detailed numerical solution or method!
