To find the factors of the quadratic equation x² – 8x + 20 = 0, we first look for numbers that multiply to the constant term (20) and add up to the linear coefficient (-8).
The constant term is 20, and we need two numbers that both multiply to 20 and add up to -8. The possible pairs of factors of 20 are (1, 20), (2, 10), (4, 5), and their negative counterparts.
However, upon examining the combinations, we notice that there are no two numbers that will both sum to -8 and multiply to 20. This suggests that the quadratic does not factor neatly into rational numbers.
We can confirm this by applying the quadratic formula: x = [ -b ± √(b² – 4ac) ] / 2a. Here, a = 1, b = -8, and c = 20.
Calculating the discriminant (b² – 4ac):
Discriminant = (-8)² – 4(1)(20) = 64 – 80 = -16.
Since the discriminant is negative, this tells us that the quadratic does not have real roots. Instead, it has complex roots.
Therefore, the factors of x² – 8x + 20 are not simple integers but rather complex numbers.
In fact, the roots of the equation can be calculated as follows:
x = [8 ± √(-16)] / 2 = [8 ± 4i] / 2 = 4 ± 2i.
From this, we can express the quadratic in its factored form involving complex numbers:
(x – (4 + 2i))(x – (4 – 2i)).
In conclusion, x² – 8x + 20 = 0 does not have real factors but can be expressed in terms of its complex roots.