1. ## Quadratic equation with imaginary roots

I have done part i of the attached question. Do I have to understand complex numbers for part ii?

2. Originally Posted by Stuck Man
I have done part i of the attached question. Do I have to understand complex numbers for part ii?

I don't see anything.

Try to post the question here or take a picture of it?

3. Originally Posted by Stuck Man
$\displaystyle \alpha = \frac{-b+\sqrt{b^2-4ac}}{2a}$

$\displaystyle \beta = \frac{-b-\sqrt{b^2-4ac}}{2a}$

4. b^2-4ac is -23. I don't know how to get a square root of a negative so thats why I'm asking if this requires a knowledge of complex numbers.

5. Originally Posted by Stuck Man
b^2-4ac is -23. I don't know how to get a square root of a negative so thats why I'm asking if this requires a knowledge of complex numbers.
Ah right, my mistake.

Yes, complex number knowledge is necessary IMO. Perhaps someone better than me can explain a way around it?

6. I've solved ii now. It takes quite a lot of writing. Square roots of -23 get squared or cancel out near the end.

7. Hello Stuck Man

• If $\displaystyle \alpha$ and $\displaystyle \beta$ are the roots of the quadratic equation $\displaystyle ax^2+bx+c=0$, then:

The sum of its roots, $\displaystyle \alpha + \beta = -\frac{b}{a}$
and
The product of its roots, $\displaystyle \alpha\beta = \frac{c}{a}$
(This is very easy to prove: expand $\displaystyle (x-\alpha)(x-\beta)$ and compare it to $\displaystyle x^2 + \frac{b}{a}x + \frac{c}{a}$.)

Well then, if $\displaystyle \alpha$ and $\displaystyle \beta$ are the roots of $\displaystyle 3x^2 +x + 2 = 0$, we get:
$\displaystyle \alpha + \beta = -\frac{1}{3}$
and
$\displaystyle \alpha\beta = \frac{2}{3}$
So for part (i):
$\displaystyle \frac{1}{\alpha^2}+\frac{1}{\beta^2} =\frac{\alpha^2+\beta^2}{\alpha^2\beta^2}$
$\displaystyle = \frac{(\alpha+\beta)^2-2\alpha\beta}{(\alpha\beta)^2}$
Now substitute in the values of $\displaystyle \alpha + \beta$ and $\displaystyle \alpha\beta$, simplify and the job's done. (I get the answer $\displaystyle -\frac{11}{4}$. Do you agree?)

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Part (ii)

We know the value of $\displaystyle \frac{1}{\alpha^2}+\frac{1}{\beta^2}$ and clearly $\displaystyle \frac{1}{\alpha^2}\times\frac{1}{\beta^2}= \frac{1}{(\alpha\beta)^2}= \frac94$. So we can use the result in reverse to write down the new equation, knowing the sum and product of its roots; i.e.
$\displaystyle x^2 - \text{(sum of roots)}x+ \text{(product of roots)}=0$
which (if my arithmetic is correct) is:
$\displaystyle x^2 +\frac{11}{4}x+\frac94=0$

i.e. $\displaystyle 4x^2+11x+9=0$

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Part (iii)

$\displaystyle \alpha$ is a root of $\displaystyle 3x^2+x+2=0$. So:
$\displaystyle 3\alpha^2+\alpha+2=0$

$\displaystyle \Rightarrow 3\alpha^2 = -\alpha-2$
(1)

$\displaystyle \Rightarrow 9\alpha^4 =(-\alpha-2)^2$
$\displaystyle =\alpha^2 +4\alpha+4$
$\displaystyle \Rightarrow 27\alpha^4 = 3\alpha^2 +12\alpha+12$
$\displaystyle =-\alpha - 2 +12 \alpha +12$ from (1)

$\displaystyle = 11\alpha+10$