Question

find a quadratic equation with the following roots .
5 +2i and 5-2i

Answer 1

\[x^2-10 x+29\]

Answer 2

how?

Answer 3

when we say a quadratic equation has roots a and b, we mean that the values a and b "satisfy" the equation. in other words, if you substitute x with either a or b, the resulting value of the function is 0. this is why roots are also called zeros

Answer 4

so if a quadratic equation has roots a and b, the quadratic equation is (x-a)(x-b)=0.
notice that if you substitute x with a, the expression becomes (a-a)(a-b) which is equal to 0.
similarly when you substitute b for x you get (b-a)(b-b) which is also equal to 0

Answer 5

(x-a)(x-b) = 0 can be expanded as
x*x -a*x-b*x +a*b =0
x^2 -(a+b)x +ab = 0
so if your roots are 5+2i and 5-2i, the resulting quadratic equation is
x^2 - (5+2i + 5-2i)x +(5+2i)(5-2i) = 0
which is
x^2 - (10)x + (5^2-(2i)^2) = 0
x^2 -10x +(25 - (-4)) =0
x^2 -10x +29 = 0