Final answer:
To solve the equation x - 1; P(x) = 3x² - 2x + 5, we need to set P(x) equal to zero and solve the quadratic equation. However, upon solving, we find that the equation has no real solutions.
Step-by-step explanation:
To solve the equation x - 1; P(x) = 3x² - 2x + 5, we need to set P(x) equal to zero, since we are looking for the values of x that make the equation true. So, we have:
3x² - 2x + 5 = 0
Now, we can solve this quadratic equation using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Comparing this formula to our equation, a = 3, b = -2, and c = 5. Substituting these values into the formula, we get:
x = (-(-2) ± √((-2)² - 4(3)(5))) / (2(3))
Simplifying further:
x = (2 ± √(4 - 60)) / 6
x = (2 ± √(-56)) / 6
Since the square root of a negative number is not a real number, there are no real solutions to this equation. It means that the equation does not have any values of x that satisfy it. Therefore, the equation x - 1; P(x) = 3x² - 2x + 5 has no solutions.