complex number Q

S

Sonal7

Full Member
Joined
Oct 4, 2019
Messages
485
z is given as 4/(1+i)

Find z, and z^2

Given that the complex root of the quadratic is( x^2)+px+q=0 where p and q are real integers, find p and q

My ans:

z can be written as 2-2i
if ax^2+bx+c is a polynomial then the the negative sum of roots is b and the product of the roots is c.

Therefore q in the equation given must be 8. This is because z^2 is the same as zz* (identity)
we know one roots which is 2 -2i therefore p=-[(2-2i)+(2+2i)], 4

Are the answers 8 and -4 correct? Is my method correct? I have no idea what the answers are to a practise sheet we were given.

Sorry to be not more helpful.
 
Sonal7 said:
z is given as 4/(1+i)

Find z, and z^2

Given that the complex root of the quadratic is ( x^2)+px+q=0 where p and q are real integers, find p and q
Click to expand...
This makes no sense as written; an equation can't be a root. I will suppose that it is intended to be,

Given that z is one complex root of the quadratic (x^2)+px+q=0 where p and q are real integers, find p and q.​

It is not true that "z^2 is the same as zz*", if you are using z* to mean the conjugate; it's z*z (using * to mean multiplication). But that isn't relevant; the important thing is that q is the product of the roots, which is indeed 8; and p is the negative of the sum of the roots, which you got correctly (eventually).

But you can check the answer, right? You are saying that the equation is x^2 - 4x + 8 = 0; you can solve that by the quadratic formula, or check by putting 2 - 2i and 2 + 2i into the equation.

(And you were very helpful, by showing your thinking.)
 
"Given that the complex root of the quadratic is( x^2)+px+q=0 where p and q are real integers, find p and q".
Is that meant to say:
"Given that z is a complex root of...."
 
Sorry! Thats exactly right Dr Peterson, it says that:

Given that z is a complex root of the quadratic equation x^2+px+q,
z=4/(1+i), find

a) z in a terms of a+bi
b) z^2

c) Find p and q.

I still think modulus z^2 is the same as zz* . I used the the ans to a and b to find p and q, and this is right you say! z squared is 8 and the product of the root and its conjugate is 8, am in right? I think a) and b) sets you up to ans c).
1576879869971.png
 
Last edited:
When you insert the word "modulus", you are right; but we can only go by what you say, not by what you mean. Be very careful when you type math!

It is simply not true that [MATH]z^2 = 8[/MATH]; rather, [MATH]z^2 = (2 - 2i)^2 = 4 - 8i + 4i^2 = -8i[/MATH].

What is true is that [MATH]|z|^2 = z\bar{z} = (2 - 2i)(2 + 2i) = 4 - 4i^2 = 8[/MATH].

I'm not sure why they asked you for [MATH]z^2[/MATH], unless to prepare you to check: [MATH]z^2 - 4z + 8 = -8i - 4(2 - 2i) + 8 = 0[/MATH]. Or possibly they expect you to know that since q is real and is equal to the product of the (conjugate) roots, it must be equal to the square of the absolute value, which is the absolute value of the square. That is, [MATH]q = z\bar{z} = |z|^2 = |z^2| = |-8i| = 8[/MATH]. I find it easier just to multiply the conjugates.

Please, please, please, when you ask about a problem, tell us the exact wording from the start, so we don't get misdirected.
 
Thank you so much! This is very helpful. It was a complete fluke that I was right about the values of p and q :)
 
Notice that \(\displaystyle \frac{1}{z}=\frac{\overline{z}}{|z|^2}\) thus \(\displaystyle \frac{4}{1+i}=2-2i\)
Now apply the complex root theorem.
 
Sonal7 said:
Sorry! Thats exactly right Dr Peterson, it says that:

Given that z is a complex root of the quadratic equation x^2+px+q,
z=4/(1+i), find
Click to expand...

Sonal7, it "says that?" "x^2 + px + q" is not a quadratic equation.
It is a quadratic expression.
 
it says z is a root of the equation x^2+px+q =0.

Sorry again was imprecise. Thanks for the correction.
 
If it was given that the coefficients, p and q, are real numbers then the two roots are z and z*. I don't see where that was given.
 
Sonal7 said:
Sorry! Thats exactly right Dr Peterson, it says that:

Given that z is a complex root of the quadratic equation x^2+px+q,
z=4/(1+i), find
a) z in a terms of a+bi
b) z^2
c) Find p and q.
Click to expand...

HallsofIvy said:
If it was given that the coefficients, p and q, are real numbers then the two roots are z and z*. I don't see where that was given.
Click to expand...

Sonal7 said:
it just said p and q are integers. That must make them real.
Click to expand...

Once again, it seems that we haven't yet seen the actual wording of the problem! In the OP you said "where p and q are real integers", but when you claimed to quote it exactly, you not only left out "= 0", but didn't mention any stated restriction on p and q.

Can you see why we request a true exact copy of the entire problem?

The easiest way is to copy exercises word-for-word (including the instructions). PLEASE use the Preview button, to check your post before submitting it. Students waste their time, when they submit posts containing typographical math errors or unreadable images.​
 
Sorry, I apologise. I thought I included the necessary details, clearly not the case. I will aim to include the exact and entire wording. Normally its a cut and paste job but this is on paper so I could not cut and paste. I would like to ask where the tool to write in mathematical notation is on this forum?

The question is exact form is as follows:

z= 4/(1+i)

Find in the form a +b i, where a, b belong to set of real number (its in symbolised form)
(a) z,
(2 marks)

(b) z^2
(2 marks)

Given that z is a complex root of the quadratic equation x^2+px+q=0, where p and q are real integers, find

(c) find the value of p and the value of q.

(3 marks)
 
Sonal7 said:
I would like to ask where the tool to write in mathematical notation is on this forum?
Click to expand...
See here. It's the calculator-like icon on the toolbar; the link provides sources for information about the language you have to use (some of which is natural, and some is not!).
 
Sonal7 said:
Sorry, I apologise. I thought I included the necessary details, clearly not the case. I will aim to include the exact and entire wording. Normally its a cut and paste job but this is on paper so I could not cut and paste. I would like to ask where the tool to write in mathematical notation is on this forum?

The question is exact form is as follows:

z= 4/(1+i)

Find in the form a +b i, where a, b belong to set of real number (its in symbolised form)
(a) z,
(2 marks)

(b) z^2
(2 marks)

Given that z is a complex root of the quadratic equation x^2+px+q=0, where p and q are real integers, find

(c) find the value of p and the value of q.

(3 marks)
Click to expand...

(a) [MATH]z=\frac{4}{1+i}\cdot\frac{1-i}{1-i}=\frac{4(1-i)}{2}=2+(-2)i[/MATH]
(b) [MATH]z^2=4-8i-4=0+(-8)i[/MATH]
(c) If \(2-2i\) is a root, then so must \(2+2i\) be a root. So, we may write:

[MATH]x^2+px+q=(x-(2-2i))(x-(2+2i))=x^2-4x+8[/MATH]
Thus, we conclude that \((p,q)=(-4,8)\).
 
Sonal7 said:
The question is exact form is as follows:

[MATH]z= 4/(1+i)[/MATH]
Find in the form [MATH]a +b i[/MATH], where a, b belong to set of real number (its in symbolised form)
(a) [MATH]z[/MATH],
(2 marks)

(b) [MATH]z^2[/MATH](2 marks)

Given that [MATH]z[/MATH] is a complex root of the quadratic equation [MATH]x^2+px+q=0[/MATH], where [MATH]p[/MATH] and [MATH]q[/MATH] are real integers, find

(c) find the value of p and the value of q.
(3 marks)
Click to expand...
Little changes in order can make a bigger difference than you think. Now I see no reason to assume that you are expected to use [MATH]z^2[/MATH] to answer (c).
 
MarkFL said:
(a) [MATH]z=\frac{4}{1+i}\cdot\frac{1-i}{1-i}=\frac{4(1-i)}{2}=2+(-2)i[/MATH]
(b) [MATH]z^2=4-8i-4=0+(-8)i[/MATH]
(c) If \(2-2i\) is a root, then so must \(2+2i\) be a root. So, we may write:

[MATH]x^2+px+q=(x-(2-2i))(x-(2+2i))=x^2-4x+8[/MATH]
Thus, we conclude that \((p,q)=(-4,8)\).
Click to expand...
Thank you! This is of great help.
 
You must log in or register to reply here.
Top