Algebra & Equations

Algebra lets us write and solve relationships between quantities. Learn variables, constants and coefficients; expressions, equations and identities; linear and simultaneous equations; quadratic equations and the discriminant; factorisation and completing the square; polynomial, rational, radical, absolute-value, exponential and logarithmic equations; and inequalities.

Almost every engineering formula is an equation to be rearranged or solved: Ohm's law for R, the projectile equation for time, the beam equation for deflection. Algebra is the toolkit for doing this reliably.

An analogy

An equation is a balance scale: whatever you do to one side, you must do to the other to keep it balanced.

Linear and simultaneous equations

Isolate the unknown with inverse operations. Two equations in two unknowns can be solved by substitution or elimination; they may have one solution, none (parallel lines) or infinitely many (same line).

Quadratics

Solve by factorising, completing the square or the formula. The discriminant tells the nature of roots. Complex roots of real quadratics come in conjugate pairs — in engineering they signal oscillation (e.g. underdamped systems).

Other equation types

Rational equations: multiply by the LCD, then reject values that make denominators zero. Radical equations: isolate and square — check for extraneous roots. Absolute-value equations split into cases. Exponential/log equations use logarithms.

Polynomials and inequalities

Polynomials of degree n have n roots (counting complex and repeated ones). Factor theorem: (x − r) is a factor if p(r) = 0. Solve inequalities with sign charts; multiplying by negatives flips the sign.

Formulas

• Quadratic equation: ax² + bx + c = 0 (a ≠ 0)
• Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)
• Discriminant: D > 0: two real roots; D = 0: one repeated root; D < 0: complex pair
• Completing the square: ax² + bx + c = a(x + b/2a)² + (c − b²/4a)
• Vieta's formulas: x₁ + x₂ = −b/a, x₁x₂ = c/a

Worked example: Quadratic formula

Given: 2x² − 3x − 5 = 0

Find: Roots.

• D = 9 + 40 = 49
• x = (3 ± 7)/4
• x = 2.5 or x = −1

Answer: x = 2.5, −1 (check: 2(6.25) − 7.5 − 5 = 0 ✓)

Worked example: Projectile time

Given: h(t) = 20t − 4.905t² m; When is h = 15 m?

Find: t.

• 4.905t² − 20t + 15 = 0
• D = 400 − 294.3 = 105.7
• t = (20 ± 10.28)/9.81

Answer: t = 0.99 s (going up) and 3.09 s (coming down)

Common mistakes

• Wrong: Dividing both sides by x (losing x = 0 root). — Right: Factor out x instead.
• Wrong: Not checking squared equations. — Right: Squaring can introduce extraneous roots — substitute back.

Applications

• Circuit design: Solving for component values.
• Kinematics: Time to reach a height or distance.
• Control & vibration: Characteristic equations s² + 2ζω_ns + ω_n² = 0.

Key takeaways

• Balance both sides.
• x = (−b ± √D)/2a; D decides root type.
• Check for extraneous roots.
• Simultaneous equations: substitution or elimination.

Going further

• Polynomial root theorems: Rational root theorem, Descartes' rule of signs.
• Cubic & quartic formulas: Cardano and Ferrari methods.
• Symbolic algebra: SymPy solve() for complex systems.