Question Which quadratic equation has roots $$ -3+6 i $$ and $$ -3-6 i ? $$ Answer $$ \begin{array}{ll} ext { Watch Video } \ x^{2}+6 x+45=0 & x^{2}-6 x+18=0\end{array} $$

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Question Which quadratic equation has roots $$ -3+6 i $$ and $$ -3-6 i ? $$ Answer $$ \begin{array}{ll}	ext { Watch Video } \ x^{2}+6 x+45=0 & x^{2}-6 x+18=0\end{array} $$

UpStudy AI · Accepted Answer

We know that if a quadratic equation has roots $$ r_1 $$ and $$ r_2 $$, then the equation can be written as

$$
(x - r_1)(x - r_2) = 0.
$$

Given the roots $$ r_1 = -3+6i $$ and $$ r_2 = -3-6i $$, we substitute them in:

$$
(x - (-3+6i))(x - (-3-6i)) = (x+3-6i)(x+3+6i) = 0.
$$

Recognize that the factors are complex conjugates. Their product can be computed using the difference of squares formula:

$$
(a-b)(a+b)=a^{2}-b^{2},
$$

with $$ a = x+3 $$ and $$ b=6i $$. Thus, we have

$$
(x+3)^2-(6i)^2 = 0.
$$

Calculate $$ (6i)^2 $$:

$$
(6i)^2 = 36i^2 = 36(-1) = -36.
$$

So the expression becomes

$$
(x+3)^2 - (-36) = (x+3)^2 + 36 = 0.
$$

Now expand $$(x+3)^2$$:

$$
(x+3)^2 = x^2+6x+9.
$$

Thus,

$$
x^2 + 6x + 9 + 36 = x^2 + 6x + 45 = 0.
$$

The quadratic equation is

$$
\boxed{x^2+6x+45=0}.
$$
The quadratic equation with roots $$ -3+6i $$ and $$ -3-6i $$ is $$ x^2 + 6x + 45 = 0 $$.

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