What is the quadratic equation forthe given roots: 4 + i and 4 - i?
For the roots (4+i) and (4-i), a quadratic equation can be written as
(x-(4+i)) (x-(4-i)) = (x-4-i) (x-4+i) = x^2 -4x-ix -4x+16+4i+xi-4i-i^2
= x^2 - 8x + 17.
Let us verify our answer by finding the roots of this equation:
x = `(8+-sqrt(8^(2) -4*17))/2 = (8+-sqrt(64-68))/2 = (8+-sqrt(-4))/2 = (8+-2i)/2 = (4+-i)`
Hence x^2 - 8x +17 is the required quadratic equation.