Final answer:
To solve for all roots of the equation x³ - 3x² + 2x = x² - 3x + 2, rearrange it into quadratic form and solve using the quadratic formula. The roots are x = 2 + √6 and x = 2 - √6.
Step-by-step explanation:
To solve for all roots of the equation x³ - 3x² + 2x = x² - 3x + 2, we rearrange it into quadratic form:
x³ - 3x² + 2x - x² + 3x - 2 = 0
x³ - 4x² + 5x - 2 = 0
Now we can solve this quadratic equation. However, it's important to note that not all quadratic equations have real solutions. But let's continue and see if we get any real roots.
Using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In our case, a = 1, b = -4, and c = -2. Plugging in these values into the formula:
x = (-(-4) ± √((-4)² - 4(1)(-2))) / (2(1))
Simplifying further:
x = (4 ± √(16 + 8)) / 2
x = (4 ± √24) / 2
x = (4 ± 2√6) / 2
x = 2 ± √6
This quadratic equation has two real roots: x = 2 + √6 and x = 2 - √6. These are the solutions for the given equation.