Final answer:
To solve 3x³ - 7x² + 3x - 7 = 0 using factor by grouping, group and factor out the common binomial (3x - 7), which gives the real solution x = 7/3.
Step-by-step explanation:
To solve the equation 3x³ - 7x² + 3x - 7 = 0 using factor by grouping, we group the terms in pairs.
Grouping the first two terms and the last two terms, we get:
(3x³ - 7x²) + (3x - 7) = 0
Next, we factor out the greatest common factor from each pair:
x²(3x - 7) + 1(3x - 7) = 0
Now, we can factor out the common binomial, (3x - 7):
(x² + 1)(3x - 7) = 0
To find the solutions, we can set each factor equal to zero:
x² + 1 = 0 or 3x - 7 = 0
The first equation, x² + 1 = 0, has no real solutions because the square of a real number cannot be negative. Thus, it has imaginary solutions.
The second equation, 3x - 7 = 0, can be solved to find the real solution:
3x = 7
x = 7/3