To solve the equation 7x^3 - x + 1 = x^3 + 3x^2 + x for real answers, you need to simplify and rearrange the equation to isolate x.
Step 1: Combine like terms on both sides of the equation:
7x^3 - x + 1 = x^3 + 3x^2 + x
Rearranging terms:
7x^3 - x^3 - 3x^2 + x - x = -1
Simplifying:
6x^3 - 3x^2 = -1
Divide both sides of the equation by 3 to simplify further:
2x^3 - x^2 = -1/3
Now the equation is in simplified form.
Step 2: To find the real solutions, you can either solve for x algebraically or graph the equation to find the x-intercepts.
To solve algebraically, you can try factoring or using numerical methods like Newton's method or the bisection method. In this case, factoring may not be straightforward.
Step 3: Another option is to graph the equation y = 2x^3 - x^2 + 1/3 and identify where the graph crosses the x-axis (y = 0). These x-values will correspond to the real solutions of the equation.
By graphing the equation, you can visually see the x-intercepts and determine the real solutions.