Question

Which of the following is equivalent to the polynomial below?

x2 – 8x + 19
A. (x + (4 + v3i)(x + (4 – v3i))
B. (r + (4 + v3i))( + - (4 – v3i))
C. (1 - (4 + v3i))( - (4 – v3i))
D. (1 + (4 – v3i)(1 + (4 – v3i))​

Answer 1

`(x-(4+√(3)i))(x-(4-√(3)i))`

Explanation:

Consider the quadratic expression

`x^2-8x+19`

First, find its discriminant:

`D=(-8)^2-4\cdot 1\cdot 19=64-76=-12`

Note that
`i^2=-1,` then
`D=12i^2.`

Use quadratic formula to find the roots:

`x_(1,2)=(-(-8)\pm √(12i^2))/(2\cdot 1)=(8\pm 2√(3) i)/(2)=4\pm √(3)i`

Now, given quadratic expression is equivalent to

`(x-(4+√(3)i))(x-(4-√(3)i))\\ \\=(x-4-√(3)i)(x-4+√(3)i)`