Final answer:
To find the complex solutions of the equation x^3 - 6x^2 + 13x - 3 = 7, rearrange the equation and use synthetic division to find the rational root. Then, solve the resulting quadratic equation using the quadratic formula to find the complex solutions x = 2 + i and x = 2 - i.
Step-by-step explanation:
To find the complex solutions of the equation x^3 - 6x^2 + 13x - 3 = 7, we first rearrange the equation to bring all terms to one side:
x^3 - 6x^2 + 13x - 3 - 7 = 0
Combining like terms, we get:
x^3 - 6x^2 + 13x - 10 = 0
Next, we can use the Rational Root Theorem or synthetic division to find the rational roots of the equation. By testing possible rational roots using synthetic division, we find that x = 2 is a root.
Dividing the cubic equation by (x - 2) using synthetic division, we get a quadratic equation:
x^2 - 4x + 5 = 0
By applying the quadratic formula, we can find the solutions for x. The quadratic equation has no real solutions because the discriminant is negative. However, it has two complex solutions:
x = 2 + i and x = 2 - i