Factoring Quadratic (imaginary numbers)

Question

Given that a, b, and c are real numbers, which of the following could NOT be the entire solution to the equation?

ax²+bx+c=0

1. x = 5i or x = 3i
2. x = 5 or x = -3
3. x = 5 + 3i
4. x = 5 - i or x = 5 + i

Answer:Of the four choices, only choice (C), x = 5 + 3i, is a root value that is not a real number .

I am trying to further my understanding of the use if imaginary numbers in this question. I am okay with factoring, but not exactly sure what they mean by root value.....isn't that just what amounts to zero?

What am i missing?

Answer 1

First of all, let me address what you said: "Of the four choices, only choice (C), x = 5 + 3i, is a root value that is not a real number."

That's incorrect.

Note that answers 1, 3 and 4 - all - have non-real solutions of the quadratic equation: they all have "i" and "i" is an imaginary/complex number.

x = 5i or x = 3i → imaginary numbers

x = 5 + 3i → imaginary numbers

x = 5 - i or x = 5 + i → imaginary numbers

Answers 1, 2 and 4 give 2 solutions.

Answer 3 gives one solution.

We know that a quadratic equation may have one of two solutions - we assume that real or complex solutions are OK.

The question was, "Which of the answers could NOT be the entire solution?"

There is only one root in answer 3, and it may not be the entire solution of the quadratic equation.

Answer 2

any quadratic equation has 2 roots or zeroes

Number 3 has only 1 root.

While it's possible the root has multiplicity 2, that never happens with an imaginary root. All imaginary roots are come in conjugate pairs, each with multiplicity one.

c: x=5+3i cannot be the entire solution. If x=5+3i is a solution, then x=5-3i is also a solution. They come in conjugate pairs

Answer 3

imaginary solutions always occur in pairs in the form of a +bi and a -bi so choice 3 could not be the entire solution