Question

Find a quadratic equation with roots -1+4i and -1-4i.

Answer 1

Hello,

(x-(-1+4i))(x-(-1-4i))=((x+1)-4i)((x+1+4i)
=((x+1)²+16
=x²+2x+17

Answer 2

`x^2 + 2x + 17`

Explanation:

Since, a quadratic equation having roots
`\alpha` and
`\beta` is,

`x^2 - (\alpha+\beta)x + \alpha.\beta`

Here,

`\alpha = -1 + 4i`

`\beta = -1 - 4i`

Hence, the required quadratic equation,

`x^2 - (-1+4i- 1 - 4i)x + (-1+4i)(-1-4i)`

`x^2 + 2x + ( (-1)^2 - (4i)^2)` ( ∵ a² - b² = (a+b)(a-b) )

`x^2 + 2x + (1 + 16)` ( ∵ i² = -1 )

`x^2 + 2x + 17`