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!