Question

Which of the following is a solution of x2 + 4x + 6?


–2 +



2 +



–2 +



2 +

Answer 1

x2 + 4x + 6 would be clearer if you'd please use the ^ symbol to indicate exponentiation:
x^2 + 4x + 6.

How would you go about finding the roots of such an equation? which of the following approaches are familiar to you?

quadratic formula
completing the square
factoring
graphing
....

Important: Determine whether the roots of x^2 + 4x + 6 are real, imaginary or complex.

Answer 2

I am familiar with factoring, but am still not even good at that. I can't even determine if the roots are real, imaginary, or complex. Could you please walk me through it? I am a virtual school student and don't get much one on one explanations.

Answer 3

Instinct tells me that the roots of this expression are complex, and that "completing the square" is one of the better ways in which to find those roots. (We can discuss the meaning of "complex" and "completing the square" later, if you wish.

Set the given x^2 + 4x + 6 = to 0 as a preliminary.

Completing the square, we re-write x^2 + 4x + 6 as x^2 + 4x + (4/2)^2 - (4/2)^2 + 6 = 0.

Now, x^2 + 4x + (4/2)^2 becomes x^2 + 4x + (2)^2, or x^2 + 4x + 4.
Next, - (4/2)^2 + 6 becomes -4+6, or -2.

Thus, your equation x^2 + 4x + (4/2)^2 - (4/2)^2 + 6 = 0.

contains a perfect square (which is x^2 + 4x + (4/2)^2) and a negative constant, -2.

Thus, your x^2 + 4x + (4/2)^2 - (4/2)^2 + 6 = 0

ends up looking like (x+2)^2 + 2 = 0, or (x+2)^2 = -2. If you find the square root of both sides, you'll get

\[x+2=\pm \sqrt{-2},\]

Answer 4

which can be re-written as\[x=-2\pm \sqrt{-2}\]

Answer 5

Can you think of a way to rewrite Sqrt(-2) using the imaginary operator, i?

I know well that this is a lot of math to swallow in one gulp! I'd welcome questions, altho' I won't be on OpenStudy for much longer tonight.

Answer 6

Looking at these roots, would you label them real, imaginary or complex? If you're not sure, it'd be a good idea to look up those words in your math book (if you have one) or online.

Examples of real numbers: 22, 5, -4

Examples of imaginary numbers: i2, i3.17, i20

Examples of complex numbers: 1+ i, 3.5 - 2i

Answer 7

Simplifying
2x + 6 = 4x + -2

Reorder the terms:
6 + 2x = 4x + -2

Reorder the terms:
6 + 2x = -2 + 4x

Solving
6 + 2x = -2 + 4x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-4x' to each side of the equation.
6 + 2x + -4x = -2 + 4x + -4x

Combine like terms: 2x + -4x = -2x
6 + -2x = -2 + 4x + -4x

Combine like terms: 4x + -4x = 0
6 + -2x = -2 + 0
6 + -2x = -2

Add '-6' to each side of the equation.
6 + -6 + -2x = -2 + -6

Combine like terms: 6 + -6 = 0
0 + -2x = -2 + -6
-2x = -2 + -6

Combine like terms: -2 + -6 = -8
-2x = -8

Divide each side by '-2'.
x = 4

Simplifying
x = 4

Answer 8

@mathmale I have taken notes on what you said, but I am confused where you got the (4/2)^2 - (4/2)^2 or even just where 4/2 came from? Could you please write out another example, I do want to understand all of this.