Final answer:
The factored expressions are (x - 3 + 2i)(x - 3 - 2i), (x + 3 + 2i)(x + 3 - 2i), (x - 4 + i)(x - 4 - i), and (x + 4 + i)(x + 4 - i).
Step-by-step explanation:
When factoring complex numbers, we can use the quadratic formula or a factoring complex numbers calculator to simplify the expression into its factors. In this case, we will be using a factoring complex numbers calculator.
To factorize the given expressions, we will follow these steps:
1. Input the expression into the factoring complex numbers calculator.
2. Click on the 'factor' button to get the factors of the expression.
3. The calculator will give the factors in the form of (a + bi)(c + di), where a, b, c, and d are coefficients.
4. Now, we need to rewrite the factors in the form of (x + a + bi)(x + a - bi), where x is the variable and a and b are the coefficients.
5. This form is called the 'conjugate form' and it is used to factorize expressions with complex numbers.
6. Comparing the given expressions with the conjugate form, we can see that the factors of (a + bi)(c + di) are (x - a + bi)(x - a - bi).
Explanation for a) (x - 3 + 2i)(x - 3 - 2i):
The factors of (x - 3 + 2i)(x - 3 - 2i) are (x + 3 + 2i)(x + 3 - 2i). This can be verified by comparing the given expression with the conjugate form. We can see that a = 3 and b = 2, so the factors will be (x - 3 + 2i)(x - 3 - 2i).
Explanation for b) (x + 3 + 2i)(x + 3 - 2i):
Similarly, the factors of (x + 3 + 2i)(x + 3 - 2i) are (x - (-3) + 2i)(x - (-3) - 2i). This can be verified by comparing the given expression with the conjugate form. We can see that a = -3 and b = 2, so the factors will be (x + 3 + 2i)(x + 3 - 2i).
Explanation for c) (x - 4 + i)(x - 4 - i):
The factors of (x - 4 + i)(x - 4 - i) are (x + 4 + i)(x + 4 - i). This can be verified by comparing the given expression with the conjugate form. We can see that a = 4 and b = 1, so the factors will be (x - 4 + i)(x - 4 - i).
Explanation for d) (x + 4 + i)(x + 4 - i):
Similarly, the factors of (x + 4 + i)(x + 4 - i) are (x - (-4) + i)(x - (-4) - i). This can be verified by comparing the given expression with the conjugate form. We can see that a = -4 and b = 1, so the factors will be (x + 4 + i)(x + 4 - i).
In conclusion, the factored expressions are (x - 3 + 2i)(x - 3 - 2i), (x + 3 + 2i)(x + 3 - 2i), (x - 4 + i)(x - 4 - i), and (x + 4 + i)(x + 4 - i). These factors can be used to simplify the original expressions and solve for the variable x.