Why the imaginary unit is needed

1. State why $x^2=-5$ has no real solution.

   2 marks
2. Simplify the principal square root $\sqrt{-36}$.

   2 marks
3. Solve $z^2+81=0$.

   2 marks
4. Solve $4z^2+64=0$.

   2 marks
5. Evaluate ${\left(3i\right)}^2$ and explain why the answer is real.

   2 marks
6. Simplify $\sqrt{-20}$ exactly, then check its square.

   3 marks

Equivalent correct methods earn the corresponding marks. Include units where asked.

1. State why $x^2=-5$ has no real solution.

   2 marks

   For $x\ge 0$ or $x<0$, multiplying $x$ by itself gives a non-negative result.

   The square of every real number is non-negative.
2. Simplify the principal square root $\sqrt{-36}$.

   2 marks

   $\sqrt{-36}=i\sqrt{36}=6i$.

   $6i$.
3. Solve $z^2+81=0$.

   2 marks

   $z^2=-81$.

   Both $9i$ and $-9i$ square to $-81$.

   $z=\pm 9i$.
4. Solve $4z^2+64=0$.

   2 marks

   $4z^2=-64$, hence $z^2=-16$.

   Take both complex roots.

   $z=\pm 4i$.
5. Evaluate ${\left(3i\right)}^2$ and explain why the answer is real.

   2 marks

   ${\left(3i\right)}^2=9i^2=9\left(-1\right)=-9$.

   The factor $i^2$ is real.

   $-9$.
6. Simplify $\sqrt{-20}$ exactly, then check its square.

   3 marks

   $\sqrt{-20}=i\sqrt{4\times 5}=2i\sqrt{5}$.

   ${\left(2i\sqrt{5}\right)}^2=4\left(-1\right)5=-20$.

   $2i\sqrt{5}$; its square is $-20$.