Question

A root of function f(x) is -1-2i. Which could be an equation for this function

f(x)=x^2-2x-3
f(x)=x^2-2x+5
f(x)=x^2+2x-3
f(x)=X^2+2x+5

Answer 1

x^2+2x+5

Answer 2

@sidsiddhartha can you please tell me how you got that?

Answer 3

you have a polynomial with only real numbers for its coefficients. That means that any complex roots come in pairs of the form \(a\pm bi\). You know one root is \(-1-2i\) so another root must be \(-1+2i\). Multiply \[(x-(-1-2i))(x-(-1+2i))\]and you've got your equation.

Answer 4

That's a bit ugly, so there's a shortcut you can do:

Say your roots are \(a\pm bi\), so the factors are \((x - a -bi)\) and \((x-a+bi)\):
Instead of multiplying \[(x-a-bi)(x-a+bi)\]regroup it a bit as \[((x-a)-bi)((x-a)+bi)\]now let \(u = x-a\) and \(v=bi\) and you have\[(u-v)(u+v)\]which is a difference of squares:
\[u^2-v^2 = (x-a)^2 - b^2i^2 = (x-a)^2 -b^2(-1) = (x-a)^2 + b^2\]You can expand that to \[(x^2-2ax+a^2) + b^2 = x^2+a^2+b^2-2ax\]which is easily evaluated:

\[x^2 -2(-1)x +(-1)^2 + (-2)^2 = x^2+2x+1+4 = x^2+2x+5\]

Answer 5

Use the discriminant. With the solution given, you need b^2-4ac to equal -16. That will give plus or minus 4i divided by two as the imaginary terms of the solutions. That means you are down to the second or fourth choices. From there, -b/2a has to be negative, so the fourth option is correct.

Answer 6

@whpalmer4
I'm sorry, this is the first time I've seen a problem like this and I really don't know what you just did...thanks though

Answer 7

That's fine, we can go back over it in smaller steps. Do you have a specific question you want me to answer, or should I just go through it with a different example that is a bit simpler?

Answer 8

Yes please...where did the u's and v's come from and the bi's??

Answer 9

Okay, the u's and v's were variables I introduced just to make it easier to see that you had a difference of squares.

A difference of squares is when you have something like \[a^2-b^2\]but the letters are unimportant — it's <one thing>^2 - <another thing>^2

So I used u to represent (x-a), and v to represent bi

Does that make sense?

Answer 10

Yes it's starting to make sense, so like is a + over - bi the formula or something that will help with other problems

Answer 11

okay, the \(a\pm bi\) is just shorthand for \(a+bi, a-bi\)

Any complex number can be written in that form. \(i = \sqrt{-1}\)

Answer 12

a number that has \(a = 0\) is called an imaginary number. a number that has \(b = 0\) is called a real number. a complex number is a combination of the two, so both \(a\) and \(b\) are not \(0\).

Answer 13

So does it get any harder than this, like I know you have to add, subtract, and simplify and graph....I'm studying for my MCA's and I haven't learned this and it's going to be on the test but I think I'm starting to pick something up

Answer 14

I don't know what MCA's are...

Answer 15

Oh they stand for Minnesota Comprehension Assessment...it's our states test, it's the second most important test next to SAT and ACT. I'm taking the math test. But yeah thanks for all your help, I may not be a total expert yet but I think I understand it a little better.

Answer 16

Ah, I'm from Minnesota too! But I'm living in sunny CA now. How did you enjoy this year's winter? I understand it may be over soon ;-)

Answer 17

California! :o sooo jealous! this year winter was painful :'( thanks for reminding...But spring is coming slowing but surely D: lols but good for you for getting outta here while you can.

Answer 18

the MCA postdates me...I left in the fall of 1981

Answer 19

even more lucky...thanks again for your help

Answer 20

do you have any more questions about this problem? or related problems?

Answer 21

No that's the only one on our practice packet....but I can't figure out this one
The surface area of a baseball is 177cm^2. What is the diameter of the baseball
Use 3.14 for pi

3.75cm
7cm
7.5cm
14.1cm

Answer 22

okay. do you know the formula for surface area of a sphere?

Answer 23

A= 4 pi r 2.

Answer 24

\[A = 4\pi r^2\]

We know \(A=177 \text{ cm}^2\)
Can you solve \[A = 4\pi r^2\] for \(r\) in terms of \(A\)?

Answer 25

I got 12.82

177=4pier^2
177=12.26 r^2
-12.56
164.44= r^2
square root 164.44 on both sides
12.82

Answer 26

No, solve the equation, then plug in numbers...

\[A = 4\pi r^2\]divide both sides by 4\[\frac{A}{4} = \pi r^2\]Divide both sides by \(\pi\)
\[\frac{A}{4\pi} = r^2\]Take square root of both sides
\[r = \sqrt{\frac{A}{4\pi}}\]Now plug in numbers...

Answer 27

by the way, it looks like you did this:
\[A = 4\pi r^2\]\[177 = 4(3.14) r^2\]\[177 = 12.56r^2\] Then you subtracted 177-12.56...can't do that!
\[177-12.56 = r^2\]

Answer 28

I got it to solve for the diameter you divide the surface area for pi....and I got 7.5 cm

Answer 29

You have to divide:
\[177 = 12.56r^2\]\[\frac{177}{12.56} = \frac{12.56r^2}{12.56} = r^2\]

Answer 30

but much better to do the problem with letters, then plug in numbers