Final answer:
To find the real solutions by factoring the equation 3x³+9=x²+27x, rearrange it into a quadratic equation and factor it by grouping. Set each factor equal to zero to find the real solutions.
Step-by-step explanation:
To find the real solutions by factoring the equation 3x³+9=x²+27x, we need to rearrange it into a quadratic equation. Subtract x² and 27x from both sides to get 3x³ - x² - 27x + 9 = 0. Next, we can factor the equation by grouping. Here's the step-by-step process:
- Factor out the GCF: x²(3x - 1) - 9(3x - 1) = 0
- Factor out the common binomial factor: (3x - 1)(x² - 9) = 0
- Apply the difference of squares formula to factor x² - 9: (3x - 1)(x - 3)(x + 3) = 0
Now, we can set each factor equal to zero to find the real solutions: 3x - 1 = 0, x - 3 = 0, and x + 3 = 0. Solving these equations gives us x = 1/3, x = 3, and x = -3. Therefore, the real solutions to the equation are x = 1/3, x = 3, and x = -3.