Question

If the roots of the equation are x = 3 ± i then the equation is x^2 + ___ + 10

Answer 1

`x^2-6x+10`

Explanation:

We know that the imaginary number i =
`√(-1)` and thus
`i^2=√(-1) √(-1) =-1`

If roots are given in the form (a+bi) and (a-bi), to find the quadratic, we can write it in the form:

(x - (a+bi) ) * (x - (a-bi))

and then multiply to figure out the answer. Shown below:

`(x-(3+i))(x-(3-i))\\=(x-3-i)(x-3+i)\\=x^2-3x+ix-3x+9-3i-ix+3i-i^2\\=x^2-6x+9-i^2\\=x^2-6x+9-(-1)\\=x^2-6x+9+1\\=x^2-6x+10`

Thus, the BLANK is -6x