What is the quadratic equation that has the roots 3+i and 3−i ?(1 point)

Question

What is the quadratic equation that has the roots  3+i  and  3−i ?(1 point)
Responses

x2+6x−10=0
x squared plus 6 x minus 10 equals 0

x2−6x+8=0
x squared minus 6 x plus 8 equals 0

x2+6x+10=0
x squared plus 6 x plus 10 equals 0

x2−6x+10=0

GPT-4o mini · Answer

To find the quadratic equation with roots $3 + i$ and $3 - i$ , we can use the fact that if $r_{1}$ and $r_{2}$ are the roots of a quadratic equation in the form $x^{2} + b x + c = 0$ , then:

The sum of the roots $r_{1} + r_{2} = - b$
The product of the roots $r_{1} \times r_{2} = c$

In this case:

$r_{1} = 3 + i$
$r_{2} = 3 - i$

Calculate the sum of the roots: $r_{1} + r_{2} = (3 + i) + (3 - i) = 3 + i + 3 - i = 6$
Calculate the product of the roots: $r_{1} \times r_{2} = (3 + i) (3 - i) = 3^{2} - i^{2} = 9 - (- 1) = 9 + 1 = 10$

Now we will use the relationships:

The sum of the roots gives us $- b = - 6$ (thus $b = - 6$ )
The product of the roots gives us $c = 10$

Therefore, the quadratic equation is: $x^{2} - 6 x + 10 = 0$

The correct response is: x²−6x+10=0 (x squared minus 6 x plus 10 equals 0)