Question

Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.The roots of an equation are x = -1 ± i. The equation is x2 + ? + 2 = 0.

Answer 1

When we know the roots are r and s we can just multiply (x-r)(x-s) and set it to zero to recover the original quadratic equation.  We can then scale to get rid of fractions or common factors if we like.

$0 = (x - (-1 + i))(x - (-1 -i)) = x^2 - ( (-1+i)+(-1-i) ) x + (-1 + i)(-1 -i)$

$0 = x^2 + 2x + 2$

The linear term coefficient is negative the sum of the roots, which in the case of conjugates is (negative) twice the real part, coefficient +2 here. The constant term is the product of the roots, which in the case of conjugates is the squared magnitude, here $1^2+1^2$.

We got a 2x in the middle so we get

Answer: 2x