To solve the equation x6 + 9x3 + 8 = 0, we can utilize a technique called u-substitution. The idea is to simplify the equation by substituting a new variable for the expression involved.
Step 1: Make the substitution
Let’s set u = x3. This means that x6 = (x3)2 = u2. Rewriting the original equation gives us:
u2 + 9u + 8 = 0.
Step 2: Factor the quadratic equation
Now we need to factor the quadratic equation. We are looking for two numbers that multiply to 8 (the constant term) and add up to 9 (the coefficient of the linear term). The numbers 1 and 8 meet these criteria:
(u + 1)(u + 8) = 0.
Step 3: Solve for u
Setting each factor equal to zero gives us:
- u + 1 = 0 ⟹ u = -1
- u + 8 = 0 ⟹ u = -8
Step 4: Substitute back for x
Now we substitute back for u using our original substitution u = x3:
- For u = -1:
x3 = -1 ⟹ x = -1. - For u = -8:
x3 = -8 ⟹ x = -2.
Step 5: Conclusion
Thus, the solutions to the equation x6 + 9x3 + 8 = 0 are:
- x = -1
- x = -2