Find $k, m$ so solutions to $(iz+k)^2=-2+2\sqrt3i$ are the same as those to $z^2-2iz+m=0$

Question

Given the equation: $(iz+k)^2=-2+2\sqrt3i$ such that the solutions of the equation are the same solution of the equation: $z^2-2iz+m=0$

Need to find $k,m$.

$z,i\in \mathbb{C}$; $k,m\in \mathbb{R}$

Got messy with the algebra.

Any nice approach here?

Thanks

You must first identify what all the constants are. Is $z$ a complex number for example? Is $k$ a real number — John, Dec 25 '13 at 14:23
$z$ is a complex of course, also $i$ such that $z=x+iy$ the rest are constants — sil, Dec 25 '13 at 14:26

John · Answer 1 · 2013-12-25T14:44:13.527

1

HINT: If you expand the LHS and compare the real and imaginary components you can find what $k$ and $z$ are. If $z$ is a complex number assume that it is equal to $a+bi$. But, the algebra might be a bit long. When you solve the second equation, you can see that $b=1$

I got $k=0$ and $m=-4$

edited Dec 25 '13 at 14:44

answered Dec 25 '13 at 14:20

John

1,149

thanks that's what i did in the first place but the algebra was too long – sil Dec 25 '13 at 14:35
It does require some extensive use of algebra but it pays of in the end. – John Dec 25 '13 at 14:39
1

Nice hint @John. :+) – Mikasa Dec 26 '13 at 09:49

score 0 · Accepted Answer · answered Dec 25 '13 at 14:22

0

HINT:

If the solutions of following two equations are the same $$ax^2+bx+c=0, Ax^2+Bx+C=0$$

then $$\frac aA=\frac bB=\frac cC$$

answered Dec 25 '13 at 14:22

lab bhattacharjee

274,582

@sil, arrange both as Quadratic Equation of $z$ and use the hint – lab bhattacharjee Dec 25 '13 at 14:29
i thought about using vieta formula but it didn't work. ill try your hint. thanks! – sil Dec 25 '13 at 14:31
@sil, yes please do that. Hope you have understood the reason behind the hint. – lab bhattacharjee Dec 25 '13 at 14:39
not really. i mean it sounds right but don't really know why – sil Dec 25 '13 at 14:40
@sil, if $\alpha,\beta$ are the roots of both the equations, what Vieta's formula says? – lab bhattacharjee Dec 25 '13 at 14:41
they will have the same sum and product by their parameters – sil Dec 25 '13 at 14:44
let us continue this discussion in chat – lab bhattacharjee Dec 25 '13 at 14:44

Find $k, m$ so solutions to $(iz+k)^2=-2+2\sqrt3i$ are the same as those to $z^2-2iz+m=0$

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