Whats the right answer
Which quadratic equation has roots negative 1 plus 4 iand negative 1 minus 4 i?.
(1 point)
Responses
x squared plus 2 x plus 2 equals 0
Image with alt text: x squared plus 2 x plus 2 equals 0
2 x squared plus x plus 17 equals 0
Image with alt text: 2 x squared plus x plus 17 equals 0
x squared plus 2 x plus 17 equals 0
Image with alt text: x squared plus 2 x plus 17 equals 0
2 x squared plus x plus 2 equals 0
The correct answer is:
x squared plus 2 x plus 17 equals 0
The right answer is:
x squared plus 2 x plus 17 equals 0
To find the quadratic equation with roots negative 1 plus 4i and negative 1 minus 4i, we need to use the fact that complex roots always come in conjugate pairs.
Let's start by representing the roots as (x - (-1 + 4i)) and (x - (-1 - 4i)).
Simplifying these expressions, we get (x + 1 - 4i) and (x + 1 + 4i), respectively.
Now, we can multiply these two expressions to get the quadratic equation:
(x + 1 - 4i)(x + 1 + 4i)
Expanding this expression using the FOIL method (First, Outer, Inner, Last), we get:
(x + 1)(x + 1) + (x + 1)(4i) - (x + 1)(4i) - (4i)(4i)
Simplifying further, we get:
(x^2 + x + x + 1) + (4ix + 4i - 4ix - 4i) - (16i^2)
Combining like terms and simplifying, we have:
x^2 + 2x + 1 + 4ix - 4ix - 16(-1)
Simplifying further, we have:
x^2 + 2x + 1 - 16 + 4ix - 4ix
Finally, combining like terms, we get:
x^2 + 2x + 1 - 16
Simplifying this expression gives us:
x^2 + 2x - 15
So, the correct quadratic equation is x^2 + 2x - 15 = 0.
Hence, none of the given options is the correct answer.