Algebra and Trigonometry: Equations, Progressions, Complex Numbers, and Triangles

Mathematics, Surveying and Transportation Engineering (MSTE) carries 35% of the CELE, and algebra and trigonometry are the gears that turn every other subject: you cannot solve a hydraulics or structural problem without them. This lesson drills the exact formulas the board expects you to reach for automatically, with the arithmetic worked all the way through so you can check your own habits against it.

Linear and quadratic equations

A linear equation isolates the variable with basic operations, always undo addition/subtraction before multiplication/division. A quadratic equation ax^2 + bx + c = 0 has roots given by the quadratic formula:

x = [-b +/- sqrt(b^2 - 4ac)] / (2a)

The discriminant, b^2 - 4ac, tells you the nature of the roots without solving: positive gives two distinct real roots, zero gives one repeated real root, negative gives two complex conjugate roots. You can also get the sum and product of roots directly: sum = -b/a, product = c/a, useful for checking an answer without re-deriving it.

Worked example: Solve 2x^2 - 7x + 3 = 0. Discriminant = (-7)^2 - 4(2)(3) = 49 - 24 = 25, so sqrt(25) = 5. x = (7 +/- 5) / 4 gives x = 3 or x = 0.5. Check: sum of roots = 3.5 = -(-7)/2, and product = 1.5 = 3/2. Both match.

Exponents and logarithms

The core exponent rules: a^m x a^n = a^(m+n); a^m / a^n = a^(m-n); (a^m)^n = a^(mn). Logarithms are the inverse operation, log_b(x) = y means b^y = x, and they obey matching laws: log(mn) = log(m) + log(n), log(m/n) = log(m) - log(n), log(m^n) = n log(m).

Worked example: Solve 2^(2x - 1) = 32. Since 32 = 2^5, match exponents: 2x - 1 = 5, so x = 3. For logarithms, given log 2 = 0.3010 and log 3 = 0.4771, the product law gives log 6 = log 2 + log 3 = 0.7781, no calculator needed once you know the individual values.

Arithmetic and geometric progressions

Type	nth term	Sum of first n terms
Arithmetic (AP): constant difference d	a_n = a1 + (n - 1)d	S_n = (n/2)[2a1 + (n - 1)d]
Geometric (GP): constant ratio r	a_n = a1 x r^(n-1)	S_n = a1(r^n - 1)/(r - 1), r not 1

For a GP with |r| < 1, the sum to infinity converges: S_infinity = a1 / (1 - r).

Worked example (AP): First term 5, common difference 3. The 10th term is a10 = 5 + (9)(3) = 32. Worked example (GP): First term 3, common ratio 2. The 6th term is a6 = 3 x 2^5 = 96, and for the series 8 + 4 + 2 + 1 + ... (a1 = 8, r = 0.5), the sum to infinity is 8 / (1 - 0.5) = 16.

Complex numbers

A complex number a + bi has i^2 = -1, and powers of i cycle every 4: i, -1, -i, 1. Multiply complex numbers the same way you multiply binomials, then simplify i^2 to -1. The modulus (magnitude) is |a + bi| = sqrt(a^2 + b^2), and the conjugate of a + bi is a - bi.

Worked example: Multiply (3 + 4i)(1 - 2i) = 3 - 6i + 4i - 8i^2 = 3 - 2i + 8 = 11 - 2i. The modulus of 3 + 4i alone is sqrt(3^2 + 4^2) = 5.

Trigonometric identities

Pythagorean identity: sin^2(theta) + cos^2(theta) = 1, which also gives 1 + tan^2(theta) = sec^2(theta) and 1 + cot^2(theta) = csc^2(theta). Double angle: sin(2theta) = 2 sin(theta) cos(theta), and cos(2theta) = cos^2(theta) - sin^2(theta).

Worked example: If sin(theta) = 3/5 in Quadrant I, then cos(theta) = sqrt(1 - 9/25) = sqrt(16/25) = 4/5, and tan(theta) = 3/4.

Laws of sines and cosines

Use the...	When you know...	Formula
Law of sines	two angles and a side (AAS, ASA), or SSA	a/sin A = b/sin B = c/sin C
Law of cosines	two sides and the included angle (SAS), or all three sides (SSS)	c^2 = a^2 + b^2 - 2ab cos C

Worked example: A triangle has a = 7, b = 8, and included angle C = 60 degrees. c^2 = 49 + 64 - 2(7)(8)(0.5) = 113 - 56 = 57, so c = sqrt(57) = 7.55.

Exam-day strategy

• Compute the discriminant first on any quadratic item; it often answers a "how many real roots" question without solving further.
• On progression problems, identify AP vs GP by checking whether consecutive terms differ by a constant amount (AP) or a constant ratio (GP).
• For complex numbers, always simplify i^2 to -1 as the very last algebraic step, doing it too early causes sign errors.
• Match the givens to the right triangle law: two sides plus the included angle or all three sides means law of cosines; an angle-side pair plus another angle or side usually means law of sines.
• Keep exact fractions (like 4/5 or 16/25) instead of decimals when the problem clearly wants a clean ratio; rounding too early is a common source of wrong answers.