Prove x² + x + 1 > 0 with Complete the Square

[thomas49th]

Homework Statement

Show x² + x + 1 > 0
well this is basically asking to prove that any value of x you put in will give a result bigger than zero, but how do i prove it

Homework Equations

According to the markscheme you should use complete the sqare

The Attempt at a Solution

(X + 1/2) + 3/4

but how does that help

Thanks :)

 

[statdad]

Can you determine whether this has any real zeros? What if it doesn't?

 

[thomas49th]

err don't see what you mean. there should be nothing to do with complex numbers in this if that's where your heading?

Thanks :)

 

[qazxsw11111]

Homework Statement

Show x² + x + 1 > 0
well this is basically asking to prove that any value of x you put in will give a result bigger than zero, but how do i prove it

Homework Equations

According to the markscheme you should use complete the sqare

The Attempt at a Solution

(X + 1/2) + 3/4

but how does that help

Thanks :)

(X + 1/2)2 + 3/4

Meaning that the graph's min pt is 3/4. Therefore it is >0.

 

[dirk_mec1]

(X + 1/2)2 + 3/4

Meaning that the graph's min pt is 3/4. Therefore it is >0.

You'll also have to show that the function is increasing.

 

[thomas49th]

to show it's increasing you'll have to have somthing > 3/4 right?


(X + 1/2)² + 3/4 = 0
x + 1/2 = sqrt(-3/4)

you can't root a minus

Is that what you have to do?

 

[statdad]

The solution has nothing to do with showing this is increasing - it can't, because x^2 + x + 1 is not an increasing function of x.


There are only a few possibilities for a quadratic:

1. It does not have any real zeros - in this case it is always positive or always negative

2. It has one real zero - in this case it is either positive and zero in one spot, or it is negative and zero in one spot

3. It has exactly two real zeros - in this case it is positive for some values of x, negative for some, and zero for 2.

So, if you can determine how many real zeros this has, and if you can determine at its sign at a value of x, you are done. Do not waste your time worrying about whether it is increasing or decreasing - that is the wrong track.

 

[Dick]

Homework Statement

Show x² + x + 1 > 0
well this is basically asking to prove that any value of x you put in will give a result bigger than zero, but how do i prove it

Homework Equations

According to the markscheme you should use complete the sqare

The Attempt at a Solution

(X + 1/2) + 3/4

but how does that help

Thanks :)

You've shown x^2+x+1=(x+1/2)^2+3/4. (x+1/2)^2 is nonnegative (it's a SQUARE). 3/4 is positive. Hence the sum is positive.

 

[Redbelly98]

(X + 1/2)2 + 3/4

Meaning that the graph's min pt is 3/4. Therefore it is >0.

Yes, absolutely. At this point, the problem is solved. The squared quantity can be zero at minimum, therefore the whole expression has a minimum of 3/4. In other words, it's always positive.

End of discussion?

 

[thomas49th]

yes thankyou very much :)