Question

Help please!
When will the equation xn = c, where c is a whole number, have two solutions?

Answer 1

When do equations in general have two solutions for one variable?

Answer 2

Do you about the quadratic function yet?

Answer 3

Yes. I do.

Answer 4

Because it can be positive or negative?

Answer 5

Yes and why is the solution positive or negative?

Answer 6

That i'm unsure about. Its been a long time since i've actually done these kind of problems.

Answer 7

It's because of the discriminant, and the fact that x^2 = (-x)^2

Answer 8

Ohhhh. Okay. That makes sense.

Answer 9

The discriminant is the part of the quadratic that discrimates between unique solutions so
what part do you know that to be?

Answer 10

x^2 = (-x)^2 is the clue

Answer 11

I'm sorry. I really don't understand this stuff. Also i'm a slow learner.

Answer 12

http://www.purplemath.com/modules/solvquad4.htm

Answer 13

This might help

Answer 14

Thank you! I appreciate it. Sorry for any inconveniences.

Answer 15

The thing to understand about this is how we can two solutions, one positive and one negative from an equation..

Answer 16

Okay. Thank you. :)

Answer 17

We can't just multiply by a number, right, how could a function say y=4x have two solutions?

Answer 18

two solutions for x?

Answer 19

it would appear there is only one x = y/4 right

Answer 20

?

Answer 21

Yes?

Answer 22

Turns out if we want two solutions, we can essentially change the magnitude of the function.. So we have x^2 = (y/4)^2 , then how would u solve for x?

Answer 23

What do we do to undo squaring a function?

Answer 24

Square root it?

Answer 25

Yes so you would get the sqrt of x^2 = the sqrt of y^2 over the sqrt of 4^2

Answer 26

What is the sqrt of x^2?

Answer 27

Would it be just x?

Answer 28

Right then what is the sqrt of 16 though?

Answer 29

4 right?

Answer 30

remember what I said before now x^2 = (-x)^2

Answer 31

Yeah

Answer 32

So 16= x^2 = (-x)^2

The sqrt of 16 is therefore not only 4

Answer 33

so its 4 and -4?

Answer 34

So we have two solutions then right? If y=4x , x = sqrt y^2 / sqrt 4 therefore x= y / 2 or x= y/-2

Answer 35

I'm confused on what you just said. Im sorry.

Answer 36

When you take the square root of a fraction by the way you'd apply the square root to both numerator and the denominator

Answer 37

Okay. So what's the question again?

Answer 38

In the problem you're dealing with now our y is just c, a whole number, like 1,2,3,4,5

Answer 39

Okayyy. So the question i asked basically is asking when will there be two solutions for c?

Answer 40

c also stands for constant many cases in math.. Say it's 4

4= nx

Answer 41

The solutions are traditionally in algebra for first of all x

Answer 42

So the question is what would n have to be in order for there to be 2 solutions for x?

Answer 43

Also in most math books n is an integer, meaning it's a number we would count on a number line but it can be positive or negative

Answer 44

n could be -5, -4,-3,-2,-1,0, 1, 2, 3, 4, 5, 6, 7, 8... or so on

Answer 45

Putting everything together we've been talking about what do you think n would have to be as a general rule?

Answer 46

I'm still confused.

Answer 47

(-x)^2 = (x)^2 sqrt of 16 is 4 and -4

Answer 48

Isn't it when x is either a positive or a negative?

Answer 49

Yes but why is it positive or negative?

Answer 50

Because it would make the answer either a positive or negative?

Answer 51

How does x become positive or negative if c=nx
What is n?

Answer 52

it's not a positive and a negative at the same time it's a positive or it's a negative in what sort of situation?

Answer 53

You said it before

Answer 54

Would it have to do with it being squared or sqrt?

Answer 55

Yes now its asking how do we get two solutions


if x= (-2)^2= 4 we have one solution

but if we have sqrt of 4

Answer 56

do you use the quadratic formula for any of this?

Answer 57

Well the quadratic tells us
-b +or- the sqrt of a discriminant all over 2 will give your solutions

Answer 58

Oh okay.

Answer 59

If the discriminant we were taking the sqrt of was 5 how many solutions would we get?

Answer 60

There would be two right?

Answer 61

i think so
one would be negative and one would be positive right?

Answer 62

There will be two solutions thanks to our discriminant

Answer 63

Yes, one discriminant would be negative and one positive

Answer 64

So what does n have to do with all of this?

Answer 65

What does this tell us about n if x=y/n

Answer 66

n is the discriminant ?

Answer 67

n is in the form of a discriminant but it isn't the discriminant because for this problem it is not necessary to use the quadratic formula

Answer 68

what is the discriminant in form of?

Answer 69

I apologize something came up.

Answer 70

That makes it yield two solutions?

Answer 71

I'm trying not to give you the answer but you are so incredibly close to it

Answer 72

If the problem is this: x^n = c, where c is a whole number,

then let n = 2 if you want 2 solutions.

x^2 = c
x = √c or x = -√c

@Taylor.Joe

Answer 73

Read up on the Fundamental Theorem of Algebra here:

https://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html

Answer 74

Gosh. Im so sorry again.

Answer 75

So what's the question again?

Answer 76

In this case,

c being a whole number = n x

what is n to yield two solutions?

You answered it back when you acknowledged there being two solutions for sqrt of 16

Answer 77

what is n in order for you to get two solutions?

Answer 78

Im so sorry dude. its x^n not xn

Answer 79

Im so sorry if you wasted your time. I feel really bad.

Answer 80

so c= x^n ? Well that should be even easier

Answer 81

Okay. Can you please explain that for me?

Answer 82

What do we know about x^2 that's so unique?

Answer 83

I will be right back and answer the question. I have to go some where real quick. I reallly appreiate your help.

Answer 84

Do you have a graphing calculator?

Answer 85

Yes

Answer 86

If you graph that thing, you'll see at point where y= 4 you will have both x=-2 and x=2

Answer 87

cause it's graph looks like a U

Answer 88

Try graphing x^3, x^4, and x^5 and x^6 and see what you notice

Answer 89

The question is to what value n must we take x so we have two solutions

Where c = x^n

Answer 90

That's two possible x values that give the same y value, no more, no less

Answer 91

Im back. I apologize.

Answer 92

What did you find about the function of x and you raise it to different values for n?

Answer 93

That there would only be two values and it would be equal to y?

Answer 94

What is y ?

x^n=c is the equation you are dealing with

Answer 95

I thought i saw y up there. I must have read something wrong.

Answer 96

y is what you would typically if in your calculator if you wanted to graph x^5 for instance

Answer 97

Ohhhhh okay