Vergelijkingen

Antwoorden bij de opgaven

1. 1. ${\displaystyle z=1}$ V ${\displaystyle z=-0.5+0.5i\sqrt{3}}$ V ${\displaystyle z=-0.5-0.5i\sqrt{3}}$
   2. ${\displaystyle z=0.5\sqrt{2}+0.5i\sqrt{2}}$ V ${\displaystyle z=0.5\sqrt{2}-0.5i\sqrt{2}}$ V ${\displaystyle z=-0.5\sqrt{2}+0.5i\sqrt{2}}$ V ${\displaystyle z=-0.5\sqrt{2}-0.5i\sqrt{2}}$
2. 1. -
   2. -
   3. ${\displaystyle z=-2.5\pm 0.5i\sqrt{15}}$
3. 1. -
   2. ${\displaystyle z_1=-1+0.5\sqrt{2}+(1+0.5\sqrt{2}i;z_2=-1cos\left(\frac{7}{12}\pi \right)+(1+sin\left(\frac{7}{12}\pi \right))i;z_3=-1+cos\left(\frac{11}{12}\pi \right)+(1+sin\left(\frac{11}{12}\pi \right))i;}$
      ${\displaystyle z_4=-1-0.5\sqrt{2}+(1-0.5\sqrt{2})i;z_5=-1+cos\left(\frac{19}{12}\pi \right)+(1+sin\left(\frac{19}{12}\pi \right))i;z_6=-1+cos\left(\frac{23}{12}\pi \right)+(1+sin\left(\frac{23}{12}\pi \right))i}$
   3. -
4. 1. ${\displaystyle z=\frac{7}{13}-\frac{22}{13}i}$
   2. ${\displaystyle z=1+2i}$
   3. ${\displaystyle z=-1.5i}$
5. 1. ${\displaystyle z_1=0.5\sqrt{3}+0.5i;z_2=-0.5\sqrt{3}+0.5i;z_3=-i}$
   2. ${\displaystyle z_1=\sqrt{2}+i\sqrt{2};z_2=-\sqrt{2}+i\sqrt{2};z_3=-\sqrt{2}-i\sqrt{2};z_4=\sqrt{2}-i\sqrt{2};}$
   3. ${\displaystyle z_1=-0.5\sqrt{2}+0.5i\sqrt{2};z_2=0.5\sqrt{2}-0.5i\sqrt{2}}$
   4. ${\displaystyle z_1=3i;z_2=-1.5\sqrt{3}-1.5i;z_3=1.5\sqrt{3}-1.5i}$
   5. ${\displaystyle z_1=\sqrt{3}+i;z_2=-2+i\sqrt{3};z_3=-\sqrt{3}-i;z_4=2-i\sqrt{3}}$
   6. ${\displaystyle z_1=0.25\sqrt{2}-0.25i\sqrt{2};z_2=-0.25\sqrt{2}+0.25i\sqrt{2}}$
6. 1. ${\displaystyle z_1=1+i;z_2=2i;z_3=-1;z_4=0}$
   2. ${\displaystyle z_1=-\frac{1}{6}+\frac{1}{6}i\sqrt{35};z_2=-\frac{1}{6}-\frac{1}{6}i\sqrt{35}}$
   3. ${\displaystyle z_1=3+i;z_2=-3-i}$
   4. ${\displaystyle z_1\approx 0.36+1.72i;z_2\approx -1.54+3.49i;z_3\approx -2.02+0.78i}$
   5. ${\displaystyle z_1=-1+0.5i\sqrt{2};z_2=-1-0.5i\sqrt{2}}$
   6. Eerst schrijven als ${\displaystyle z^4=-16}$ V ${\displaystyle z^4=1}$ etc.
7. 1. ${\displaystyle z=3.2-2.4i}$
   2. ${\displaystyle z_1=1+\sqrt{2}+i\sqrt{2};z_2=1-\sqrt{2}-i\sqrt{2}}$
   3. ${\displaystyle z_1=1.5+0.5i\sqrt{3};z_2=i\sqrt{3};z_3=-1.5+0.5i\sqrt{3};z_4=-1.5-0.5i\sqrt{3};z_5=-i\sqrt{3};z_6=1.5-0.5i\sqrt{3}}$
   4. ${\displaystyle z=0}$
8. ${\displaystyle z=\pm \sqrt{e^{-0.5\pi }}}$
9. 1. -
   2. ${\displaystyle x=2}$
   3. ${\displaystyle x^3+6x-20=(x-2)(x^2+2x+10)=0}$ geeft ${\displaystyle x=2}$ V ${\displaystyle x=-1\pm 3i}$
   4. ${\displaystyle x=-4}$ V ${\displaystyle x=-0.5\pm 0.5i\sqrt{3}}$
   5. ${\displaystyle x=\frac{1}{3}}$ V ${\displaystyle x=-2\pm i\sqrt{2}}$
10. 1. ${\displaystyle z=0}$ V ${\displaystyle z=\pm 0.5\sqrt{3}+1.5i}$
    2. ${\displaystyle z=0.4+0.2i}$ V ${\displaystyle z=-0.4-0.2i}$
    3. ${\displaystyle z_1\approx 1.93+0.52i;z_2\approx 0.52+1.93i;z_3\approx -1.41+1.41i;z_4\approx -1.93-0.52i;z_5\approx -0.52-1.93i;z_6\approx 1.41-1.41i}$
    4. ${\displaystyle z_1\approx -0.62-3.12i;z_2\approx 0.12+0.62i}$