Algebra

This chapter covers the following topics:

• Quadratic equation
• Logarithms
• Complex numbers
• Progression and series

These formulas all appear in the Mathematics section of the NCEES FE Reference Handbook (under “Algebra and trigonometry,” “Complex numbers,” and “Progressions and series”). You don’t need to memorize them - practice finding and applying them quickly.

Quadratic equation

A quadratic equation is a second-degree polynomial equation written in the form:

$$a{x}^2+bx+c=0$$

where $a,b,c$ are constants, $a\ne 0$, and $x$ is the variable.

You can find the roots (solutions) of a quadratic equation using the quadratic formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Discriminant ($\Delta$)

The expression under the square root is called the discriminant:

$$\Delta =b^2-4ac$$

The discriminant tells you what kind of roots the equation has:

If $\Delta >0$: the equation has two distinct real roots.

If $\Delta =0$: the equation has one real root (a repeated root).

If $\Delta <0$: the equation has two complex conjugate roots.

Example: solving a quadratic equation

Solve $x^2-5x+6=0$.

• $a=1$, $b=-5$, $c=6$, so $\Delta =25-24=1>0$ (two real roots).
• $x=\frac{5\pm 1}{2}$

Answer: $x=2$ or $x=3$

Sum and product of roots

For a quadratic equation $a{x}^2+bx+c=0$, the sum and product of its roots are:

Sum of roots:

$$r_1+r_2=-\frac{b}{a}$$

Product of roots:

$$r_1{r}_2=\frac{c}{a}$$

Logarithms

The logarithm of $x$ to the base $b$ is defined by:

$${\log }_b(x)=c$$

where:

$$b^c=x$$

Special definitions for $b=e$ or $b=10$ are:

$$\ln x(b=e)$$

$$\log x(b=10)$$

To change from one base to another, use the change-of-base formula:

$${\log }_{b}x=\frac{{\log }_{a}x}{{\log }_{a}b}$$

e.g.,

$$\ln x=\frac{{\log }_{10}x}{{\log }_{10}e}=2.302585\left({\log }_{10}x\right)$$

Exam tip: approved exam calculators compute $\log$ (base $10$) and $\ln$ (base $e$) directly, but not other bases. Use the change-of-base formula above for expressions like ${\log }_{3}x$, and keep the conversion factor (e.g., $2.302585$) at full precision rather than rounding mid-problem.

Identities

$${\log }_b{b}^n=n$$

$$\log x^c=c\log x,x^c=\text{antilog}(c\log x)$$

$$\log (xy)=\log x+\log y$$

$${\log }_{b}b=1,\log 1=0$$

$$\log \frac{x}{y}=\log x-\log y$$

Example: evaluating a logarithmic expression

Evaluate ${\log }_{2}32+{\log }_{2}4$.

• ${\log }_{2}32+{\log }_{2}4={\log }_2(32\times 4)={\log }_{2}128={\log }_2{2}^7=7$

Answer: $7$

Complex numbers

Definitions

Complex number

Written as $z=a+bi$, where $a$ is the real part ($Re(z)$), $b$ is the imaginary part ($Im(z)$), and $i$ is the imaginary unit satisfying $i^2=-1$.

Conjugate

The conjugate of $z=a+bi$ is $\bar{z}=a-bi$.

Modulus

The modulus (magnitude) of $z=a+bi$ is $|z|=\sqrt{a^2+b^2}$.

Argument

The angle $\theta$ that $z$ makes with the positive real axis: $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$, adding $\pi$ to the result when $a<0$.

Watch out: when $a<0$, ${\tan }^{-1}(b/a)$ lands in the wrong quadrant - add $\pi$ and check the sign of $b$ to confirm the quadrant. Also avoid rounding $r$ or $\theta$ before converting back to rectangular form; that’s a common source of error.

Operations with complex numbers

Addition and subtraction

For two complex numbers $z_1=a+bi$ and $z_2=c+di$:

$$z_1+z_2=(a+c)+(b+d)i$$

$$z_1-z_2=(a-c)+(b-d)i$$

Multiplication

$$z_1\cdot z_2=(a+bi)(c+di)=(ac-bd)+(ad+bc)i$$

Division

$$\frac{z_1}{z_2}=\frac{(a+bi)(c-di)}{c^2+d^2}=\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i$$

Polar form of a complex number

A complex number can also be represented in polar form:

$$z=r(\cos \theta +i\sin \theta )$$

where $r=|z|=\sqrt{a^2+b^2}$ is the modulus and $\theta$ is the argument of $z$.

Euler’s formula and identity

Euler’s formula states:

$$e^{i\theta }=\cos \theta +i\sin \theta$$

So the polar form can also be written as:

$$z=r{e}^{i\theta }$$

Euler’s identity

A special case of Euler’s formula occurs when $\theta =\pi$:

$$e^{i\pi }+1=0$$

This is known as

Euler’s identity.

De Moivre’s theorem

For any integer $n$, De Moivre’s theorem states:

$$(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta )$$

or equivalently in exponential form:

$${\left(e^{i\theta }\right)}^n=e^{in\theta }$$

Example: computing $z^4$ with De Moivre’s theorem

Find $z^4$ for $z=1+i$.

• $r=\sqrt{1^2+1^2}=\sqrt{2}$, $\theta ={\tan }^{-1}(1)=\frac{\pi }{4}$ (no quadrant adjustment since $a>0$).
• $z^4=r^4(\cos 4\theta +i\sin 4\theta )=4(\cos \pi +i\sin \pi )=4(-1+0i)$

Answer: $z^4=-4$

Progression and series

Arithmetic progression

To check whether a finite sequence is an arithmetic progression, subtract each term from the next term. If the differences are equal, the sequence is arithmetic.

1. The first term is $a$.
2. The common difference is $d$.
3. The number of terms is $n$.
4. The last or $n$th term is $l$.
5. The sum of $n$ terms is $S$.

$$l=a+(n-1)d$$

$$S=\frac{n(a+l)}{2}=\frac{n\left[2a+(n-1)d\right]}{2}$$

Geometric progression

To check whether a finite sequence is a geometric progression (G.P.), divide each term after the first by the preceding term. If the quotients are equal, the sequence is geometric.

1. The first term is $a$.
2. The common ratio is $r$.
3. The number of terms is $n$.
4. The last or nth term is $l$.
5. The sum of $n$ terms is $S$.

$$l=a{r}^{(n-1)}$$

$$S=\frac{a(1-r^n)}{1-r},r\ne 1$$

$$S=\frac{a-rl}{1-r},r\ne 1$$

$$S_{\infty }=\frac{a}{1-r},|r|<1$$

A G.P. converges if $|r|<1$ and it diverges if $|r|>1$.

Example: finding the first term of a geometric progression

The 2nd term of a G.P. is $6$ and the 6th term is $486$. Find $a$.

• $ar=6$ and $a{r}^5=486$, so $r^4=81$, giving $r=3$ (keep this exact, not rounded).
• $a(3)=6$, so $a=2$.

Answer: $a=2$