Complex Numbers and Quadratic Equations

The One Thing

Complex numbers extend the real number system by defining $i^2=-1$, allowing equations such as $x^2+1=0$ to have solutions. Quadratic equations can then be solved systematically using factorisation, completing the square, the quadratic formula, and the discriminant, including cases with complex roots.

Definitions and Results

• Imaginary unit: $i^2=-1$, so $i=\sqrt{-1}$.
• Complex number: A number of the form $z=a+ib$, where $a$ and $b$ are real numbers.
• Real part: For $z=a+ib$, $\mathrm{Re}(z)=a$.
• Imaginary part: For $z=a+ib$, $\mathrm{Im}(z)=b$, not $ib$.
• Purely real number: A complex number $a+0i$, whose imaginary part is zero.
• Purely imaginary number: A complex number $0+ib$, where $b$ is real.
• Equality of complex numbers: $a+ib=c+id$ if and only if $a=c$ and $b=d$.
• Conjugate: If $z=a+ib$, its conjugate is $\overline{z}=a-ib$.
• Modulus: $|z|=\sqrt{a^2+b^2}$, representing the distance from $z$ to the origin.
• Argand plane: A coordinate plane in which the horizontal axis represents real parts and the vertical axis represents imaginary parts.
• Complex number as an ordered pair: $a+ib$ corresponds to the point $(a,b)$ in the Argand plane. Its conjugate corresponds to $(a,-b)$.
• Powers of $i$: The powers repeat in a cycle of four:

$$i^0=1,i^1=i,i^2=-1,i^3=-i,i^4=1.$$

• Addition and subtraction:

$$(a+ib)\pm (c+id)=(a\pm c)+i(b\pm d).$$

• Multiplication:

$$(a+ib)(c+id)=(ac-bd)+i(ad+bc),$$ using $i^2=-1$.

• Conjugate product:

$$z\overline{z}=(a+ib)(a-ib)=a^2+b^2=|z{|}^2.$$

• Division: To divide by a non-zero complex number, multiply the numerator and denominator by the conjugate of the denominator.
• Modulus of a conjugate product:

$$|z\overline{z}|=|z{|}^2.$$

• Distance in the Argand plane: For $z_1=a+ib$ and $z_2=c+id$,

$$|z_1-z_2|=\sqrt{(a-c{)}^2+(b-d{)}^2}.$$

• Quadratic equation: An equation of the form

$$a{x}^2+bx+c=0,$$ where $a,b,c$ are real numbers and $a\ne 0$.

• Roots or zeros: Values of $x$ that satisfy the quadratic equation.
• Discriminant: For $a{x}^2+bx+c=0$,

$$D=b^2-4ac.$$ It determines the type of roots: - If $D>0$, there are two distinct real roots. - If $D=0$, there are two equal real roots: $$x=-\frac{b}{2a}.$$ - If $D<0$, there are two non-real complex conjugate roots.

• Quadratic formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a},a\ne 0.$$ When $D<0$, this may be written as $$x=\frac{-b\pm i\sqrt{-D}}{2a}.$$

• Vieta’s relations: If $\alpha$ and $\beta$ are roots of $a{x}^2+bx+c=0$, then

$$\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}.$$ Conversely, a quadratic with roots $\alpha$ and $\beta$ is $$x^2-(\alpha +\beta )x+\alpha \beta =0.$$

• Complex conjugate roots: For real coefficients, non-real roots always occur in conjugate pairs. Thus, if $a+ib$ is one root, $a-ib$ is the other.

Worked Methods

1. Simplifying powers of $i$

Use the cycle $$i^0=1,i^1=i,i^2=-1,i^3=-i,i^4=1.$$

For higher powers, reduce the exponent modulo $4$. For example, powers differing by $4$ have the same value because $i^4=1$.

2. Adding and subtracting complex numbers

Given $$(a+ib)\pm (c+id),$$ combine real parts with real parts and imaginary parts with imaginary parts: $$(a+ib)\pm (c+id)=(a\pm c)+i(b\pm d).$$

Equality or any resulting equation must be checked by comparing real and imaginary parts separately.

3. Multiplying complex numbers

Expand as ordinary algebra: $$(a+ib)(c+id)=ac+aid+bic+b{i}^{2}d.$$

Use $i^2=-1$: $$ac+iad+ibc-bd=(ac-bd)+i(ad+bc).$$

Thus, $$(a+ib)(c+id)=(ac-bd)+i(ad+bc).$$

4. Finding a conjugate and modulus

For $$z=a+ib,$$ change the sign of the imaginary term to obtain $$\overline{z}=a-ib.$$

Then $$z\overline{z}=(a+ib)(a-ib)=a^2+b^2,$$ and therefore $$|z|=\sqrt{a^2+b^2},|z{|}^2=a^2+b^2.$$

5. Dividing complex numbers

To divide by a non-zero complex denominator, multiply the numerator and denominator by the denominator’s conjugate.

For a quotient with denominator $c+id$, use $$\frac{a+ib}{c+id}\times \frac{c-id}{c-id}.$$

The denominator becomes $$(c+id)(c-id)=c^2+d^2,$$ which is real and non-zero. Expand the numerator and write the result in the form $p+iq$.

6. Representing complex numbers geometrically

For $$z=a+ib,$$ plot the point $$(a,b)$$ in the Argand plane. The horizontal coordinate is the real part and the vertical coordinate is the imaginary part.

The conjugate $$\overline{z}=a-ib$$ is represented by $$(a,-b),$$ which is the reflection of $z$ across the real axis.

The modulus $$|z|=\sqrt{a^2+b^2}$$ is the distance from the origin. For two complex numbers, $$z_1=a+ib,z_2=c+id,$$ their distance is $$|z_1-z_2|=\sqrt{(a-c{)}^2+(b-d{)}^2}.$$

7. Solving a quadratic by factorisation

For a quadratic equation, first rearrange it into the form $$a{x}^2+bx+c=0.$$

When suitable factors are easily identified, factorise the quadratic. Set each factor equal to zero and solve the resulting linear equations. This method is most direct when the quadratic has simple factors.

8. Solving a quadratic by completing the square

Rearrange the quadratic so that the $x^2$ and $x$ terms are together. Add and subtract the required constant to form a perfect square: $$x^2+px={\left(x+\frac{p}{2}\right)}^2-\frac{p^2}{4}.$$

The resulting squared form directly reveals the roots or can be used to derive the quadratic formula.

9. Solving a quadratic using the quadratic formula

For $$a{x}^2+bx+c=0,a\ne 0,$$ identify $a$, $b$, and $c$, then substitute into $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

Evaluate the discriminant $$D=b^2-4ac$$ to determine the form of the roots. If $D<0$, use $$\sqrt{D}=i\sqrt{-D},$$ giving $$x=\frac{-b\pm i\sqrt{-D}}{2a}.$$

10. Using the discriminant to classify roots

For $$a{x}^2+bx+c=0,$$ calculate $$D=b^2-4ac$$ without necessarily solving the equation:

• $D>0$: two distinct real roots.
• $D=0$: two equal real roots, $x=-b/(2a)$.
• $D<0$: two non-real complex conjugate roots.

11. Using Vieta’s relations

If the roots are $\alpha$ and $\beta$, use $$\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}.$$

To construct a monic quadratic from the roots, use $$x^2-(\alpha +\beta )x+\alpha \beta =0.$$

For a non-monic quadratic, multiply the equation by the required non-zero leading coefficient.

Where It Goes Wrong

• Confusing the imaginary part $b$ with the imaginary term $ib$; for $z=a+ib$, $\mathrm{Im}(z)=b$.
• Forgetting to use $i^2=-1$ when multiplying complex numbers, or failing to reduce powers of $i$ using their cycle of four.
• Omitting the conjugate when dividing by a complex number, leaving a complex denominator rather than a real denominator.
• Comparing complex numbers without comparing real parts and imaginary parts separately.
• Using the discriminant classification without checking that the equation is in the form $a{x}^2+bx+c=0$ with $a\ne 0$.
• Forgetting that when $D<0$, the square root is written as $i\sqrt{-D}$, and that real-coefficient quadratics then have conjugate roots.

What Gets Asked

• Define $i$, complex numbers, real and imaginary parts, conjugates, moduli, and the Argand plane.
• Simplify expressions involving powers of $i$.
• Add, subtract, multiply, and divide complex numbers.
• Find a complex conjugate and calculate $z\overline{z}$, $|z|$, or $|z\overline{z}|$.
• Represent $a+ib$ and $a-ib$ in the Argand plane.
• Calculate the distance between two complex numbers using

$$|z_1-z_2|=\sqrt{(a-c{)}^2+(b-d{)}^2}.$$

• Solve quadratic equations by factorisation, completing the square, or the quadratic formula.
• Use the discriminant $D=b^2-4ac$ to classify roots.
• Express non-real roots in the form

$$\frac{-b\pm i\sqrt{-D}}{2a}.$$

• Use Vieta’s relations to find the sum or product of roots.
• Form a quadratic equation from prescribed roots $\alpha$ and $\beta$.
• Explain why non-real roots of a quadratic with real coefficients occur in conjugate pairs.