# Which quadratic equation has roots -1+4i and -1-4i ?

A. x^2+2x+2=0
B. 2x^2+x+17=0
C. x^2+2x+17=0
D. 2x^2+x+2=0

I'm having difficulty with this question, but I've settled on C for now. Am I correct?

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B1 3/2
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## (x + 1 - 4i)(x + 1 + 4i)

x^2 + x + 4ix + x + 1 + 4i - 4ix - 4i - (4i)^2

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## −16i^2+x^2+2x+1

But that's not one of my possible answers?

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## To determine which quadratic equation has the given roots, you can use the fact that complex roots of a quadratic equation always occur in conjugate pairs.

First, let's write the general form of a quadratic equation:
ax^2 + bx + c = 0

The roots of the quadratic equation can be found using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a

You have the roots -1+4i and -1-4i. Notice that the real part (-1) remains the same for both solutions, while the imaginary part (±4i) changes the sign. This means the equation has a complex conjugate pair of roots.

-1+4i = (-b + √(b^2 - 4ac)) / 2a

From this equation, we can see that the discriminant (b^2 - 4ac) must be negative to account for the complex roots. Now, let's focus on the second root -1-4i:
-1-4i = (-b - √(b^2 - 4ac)) / 2a

Again, the discriminant must be negative here as well. By comparing these two equations, we deduce that the discriminant must be a negative value, which eliminates option B and D.

Now, let's analyze options A and C:
A. x^2 + 2x + 2 = 0
C. x^2 + 2x + 17 = 0

To further narrow down and confirm the correct choice, we can compare the coefficients of the quadratic equations. In particular, focus on the middle term (bx) since both roots share -2 as their real part (which means the sum of the roots should be -2). In option A, we have 2x, while in option C, we have 2x. This confirms that option A is not correct, as its middle term differs from the desired -2x.

Therefore, the correct answer is option C:
C. x^2 + 2x + 17 = 0

You are correct! Well done.