The Argand Plane and Quadratic Equations: Quick Revision

A brief note on The Argand Plane and Quadratic Equations. Read it before a test. Try each question at the end before you open its answer.

Conjugate and modulus

• For $z = a + ib$, the conjugate is $\bar z = a - ib$ and the modulus is $|z| = \sqrt{a^2 + b^2}$.
• $z\bar z = |z|^2$, $z + \bar z = 2a$ and $z - \bar z = 2ib$.
• $\overline{z_1 + z_2} = \bar z_1 + \bar z_2$ and $\overline{z_1 z_2} = \bar z_1 \bar z_2$.
• $|z_1 z_2| = |z_1||z_2|$ and $\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}$.
• $|z_1 + z_2| \le |z_1| + |z_2|$, usually with a strict inequality.

Dividing

• Multiply the top and the bottom by the conjugate of the bottom: $\dfrac{z_1}{z_2} = \dfrac{z_1 \bar z_2}{|z_2|^2}$.
• The inverse of $z \ne 0$ is $z^{-1} = \dfrac{\bar z}{|z|^2}$.

The Argand plane

Quadratic equations

• For $ax^2 + bx + c = 0$ with real coefficients and $D = b^2 - 4ac < 0$, the roots are $\dfrac{-b \pm i\sqrt{-D}}{2a}$.
• The roots are a conjugate pair, mirror images in the real axis.
• Their sum is $-\dfrac{b}{a}$, and their product is $\dfrac{c}{a} = |\alpha|^2$.

Square roots

To find $\sqrt{a + ib}$, write $(x + iy)^2 = a + ib$. Then:

• $x^2 - y^2 = a$ and $2xy = b$;
• $x^2 + y^2 = \sqrt{a^2 + b^2}$;
• $x$ and $y$ have the same sign when $b > 0$, and opposite signs when $b < 0$.

A worked example

Solve $x^2 - 6x + 34 = 0$.

Step 1. The discriminant. $D = 36 - 136 = -100$.

Step 2. The roots. $x = \dfrac{6 \pm 10i}{2} = 3 \pm 5i$.

Step 3. Check. The sum is $6$, and the product is $9 + 25 = 34$.

Practice with new numbers

A program to try

The program solves the worked example with cmath and checks that the roots are conjugates with equal moduli.

Mistakes to avoid

• The wrong sign in the conjugate. Only the imaginary part changes sign.
• $|a + ib|$ as $a + b$. It is $\sqrt{a^2 + b^2}$.
• Distance as $|z_1| - |z_2|$. It is $|z_1 - z_2|$.
• No solution for a negative $D$. There are two complex roots.
• $\sqrt{-36}$ as $-6$. It is $6i$.

Check yourself

1. Write $\dfrac{4 + 3i}{2 - i}$ in the form $a + ib$.

2. What is the modulus of $(1 + i)^6$?

3. The points $z$ and $\bar z$ are $8$ apart. What is the imaginary part of $z$, if it is positive?

4. Solve $x^2 + 6x + 13 = 0$.

5. Find the square roots of $-21 - 20i$.