Questions: Find all real solutions of the equation. (Enter your answers as x^3-3 x^2+x-3=x^2+1

Solution Steps

To find the real solutions of the given equation, we first need to simplify it by moving all terms to one side, resulting in a polynomial equation. Then, we can use numerical methods or a root-finding algorithm to find the real roots of the polynomial.

The given equation is: \[ x^3 - 3x^2 + x - 3 = x^2 + 1 \] First, move all terms to one side to form a polynomial equation: \[ x^3 - 3x^2 + x - 3 - x^2 - 1 = 0 \] Simplify the equation: \[ x^3 - 4x^2 + x - 4 = 0 \]

To find the real solutions, we solve the polynomial equation: \[ x^3 - 4x^2 + x - 4 = 0 \] The roots of this polynomial are: \[ x \approx 4.2242, \quad x \approx -0.1121 + 0.9666i, \quad x \approx -0.1121 - 0.9666i \]

Among the roots, only the real number is a valid solution. The complex roots are not considered as real solutions.

Final Answer