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?
B-6-2
A7-
B1 3/2
D
By=
C7.5
A 1
D
C
x's answers to the quiz are correct
it’s c :)
(x + 1 - 4i)(x + 1 + 4i)
x^2 + x + 4ix + x + 1 + 4i - 4ix - 4i - (4i)^2
−16i^2+x^2+2x+1
But that's not one of my possible answers?
i = √-1 ... i^2 = -1
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.
To find the quadratic equation, we substitute these roots into the quadratic formula. Let's start with the root -1+4i:
-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.