Another quadratic question: For equation x^2 = 2x -3 with root p show that p^3 = p-6

[tangaloomaflyer]

OK, here's another one:

For equation x² = 2x -3 with root p

show that p³ = p-6

So I've re-arranged to get x² -2x +3

The roots are complex: 1+(i*sqrt2) and 1-(i*sqrt2)

I can't see how p³ = p-6

For p³ i get:
-1+(i*sqrt2) or 3-(3*i*sqrt2)

For p-6 I get:
(i*sqrt2) - 5 or (i*sqrt2) -4

Where to go from here?

 

[stapel]

For equation x² = 2x -3 with root p, show that p³ = p-6

So I've re-arranged to get x² -2x +3

The roots are complex: 1+(i*sqrt2) and 1-(i*sqrt2)

I can't see how p³ = p-6

For p³ i get:
-1+(i*sqrt2) or 3-(3*i*sqrt2)

I'm not getting these values...? For instance:

. . . . .\(\displaystyle \left(1\, +\, \sqrt{\strut 2\,}\, i\right)\, \left(1\, +\, \sqrt{\strut 2\,}\, i\right)\, =\, 1\, +\, 2\, \sqrt{\strut 2\,}\,i\, -\, 2\, =\, -1\, +\, 2\, \sqrt{\strut 2\,}\, i\)

. . . . .\(\displaystyle \begin{align}\left(1\, +\, \sqrt{\strut 2\,}\, i\right)\, \left(-1\, +\, 2\, \sqrt{\strut 2\,}\, i\right)\, &=\, -1\, -\, \sqrt{\strut 2\,}\, i\, +\, 2\, \sqrt{\strut 2\,}\, i\, -\, 2\, \cdot\, 2\,

\\ \\ &=\, -1\, +\, \sqrt{\strut 2\,}\, i\, -\, 4\, =\, -5\, +\, \sqrt{\strut 2\,}\, i\end{align}\)

 

[pka]

OK, here's another one:

For equation x² = 2x -3 with root p

show that p³ = p-6

Let \(\displaystyle \bf{p}=1+\sqrt2\bf{i}\) then \(\displaystyle \bf{p^3=-5+i\sqrt 2=(1+i\sqrt 2)-6=p-6}\)

 

[tangaloomaflyer]

I'm not getting these values...? For instance:

. . . . .\(\displaystyle \left(1\, +\, \sqrt{\strut 2\,}\, i\right)\, \left(1\, +\, \sqrt{\strut 2\,}\, i\right)\, =\, 1\, +\, 2\, \sqrt{\strut 2\,}\,i\, -\, 2\, =\, -1\, +\, 2\, \sqrt{\strut 2\,}\, i\)

. . . . .\(\displaystyle \begin{align}\left(1\, +\, \sqrt{\strut 2\,}\, i\right)\, \left(-1\, +\, 2\, \sqrt{\strut 2\,}\, i\right)\, &=\, -1\, -\, \sqrt{\strut 2\,}\, i\, +\, 2\, \sqrt{\strut 2\,}\, i\, -\, 2\, \cdot\, 2\,

\\ \\ &=\, -1\, +\, \sqrt{\strut 2\,}\, i\, -\, 4\, =\, -5\, +\, \sqrt{\strut 2\,}\, i\end{align}\)

Yep, thanks - totally messed up at this point - I see where I went wrong

 

[tangaloomaflyer]

You mean: x^2 - 2x + 3 = 0

(in case your teacher deducts a point or 2!)

Schoolboy error!

(I'm not actually studying - I'm 36 but I never really learned this first time around, so I'm re-teaching myself from my old A-level textbook)