leaving cert complex numbers hl

mayogirlie1

Anyone please help quickly with 2nd part!

TheBody

From the first part you should have [latex]z^2=-2-2\sqrt{3}i[/latex]

If we sub the vaules for [latex]z^2[/latex] and [latex]z[/latex] into the statement in the second part we get:

[latex]z^2+pz=(-2-2\sqrt{3}i)+p(-1+\sqrt{3}i)[/latex].

When we multiply out the brackets and simplify we get:

[latex]z^2+pz=(-2-p)+i(-2\sqrt{3}+p\sqrt{3})[/latex].

As we want this to be a real number, that means we want the complex part to be zero. So we just need to let:

[latex]-2\sqrt{3}+p\sqrt{3}=0[/latex]

Solving for [latex]p[/latex] you will get that [latex]p=2[/latex].

I'll let you fill in the calculation details. If you are still having problems, let me know and I'll provide some more detail.

Good luck!