Whestion Which quadratic equation has roots $$ -4+2 i $$ and $$ -4-2 i $$ ? Answer $$ \begin{array}{ll}3 x^{2}-24 x+60=0 & 3 x^{2}+24 x+69=0 \ 3 x^{2}-24 x+69=0 & 3 x^{2}+24 x+60=0\end{array} $$

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1. Let the quadratic equation with roots $$ r_1 = -4+2i $$ and $$ r_2 = -4-2i $$ be written in the form
   $$
   x^2 - (r_1 + r_2)x + r_1 r_2 = 0.
   $$

2. First, compute the sum of the roots:
   $$
   r_1 + r_2 = (-4+2i) + (-4-2i) = -8.
   $$
   Thus, the coefficient of $$ x $$ is
   $$
   -(r_1 + r_2) = -(-8) = 8.
   $$

3. Next, compute the product of the roots:
   $$
   r_1 r_2 = (-4+2i)(-4-2i).
   $$
   Using the formula for the product of a complex number and its conjugate:
   $$
   (-4+2i)(-4-2i) = (-4)^2 - (2i)^2 = 16 - (4i^2).
   $$
   Since $$ i^2 = -1 $$, we have:
   $$
   16 - (4 \cdot -1) = 16 + 4 = 20.
   $$

4. Substituting the computed sum and product into the quadratic equation:
   $$
   x^2 + 8x + 20 = 0.
   $$

5. The problem asks for an equivalent equation of the form with a leading coefficient of $$ 3 $$. Multiply the entire equation by $$ 3 $$:
   $$
   3x^2 + 24x + 60 = 0.
   $$

Thus, the quadratic equation with roots $$ -4+2i $$ and $$ -4-2i $$ is 
$$
3x^2 + 24x + 60 = 0.
$$
The quadratic equation with roots $$ -4+2i $$ and $$ -4-2i $$ is $$ 3x^2 + 24x + 60 = 0 $$.

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Which quadratic equation has roots and ?
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Whestion Which quadratic equation has roots and ? Answer