General form
ax + b = 0. Solution: x = −b/a if a ≠ 0.
Loading page
Resource 1 of 2
Lesson overview
This lesson brings together the PDF, study notes and chapter flow so you can revise Equations & Systems without losing the thread.
General form
ax + b = 0. Solution: x = −b/a if a ≠ 0.
1st degree inequality
ax + b > 0: if a > 0, x > −b/a; if a < 0, x < −b/a.
General form
ax² + bx + c = 0 with a ≠ 0.
Discriminant Δ
Δ = b² − 4ac. Δ > 0: 2 roots | Δ = 0: 1 double root | Δ < 0: no real roots.
Root formulas
x₁ = (−b − √Δ) / (2a) and x₂ = (−b + √Δ) / (2a)
Vieta's formulas
x₁ + x₂ = −b/a and x₁ · x₂ = c/a
Factorization
ax² + bx + c = a(x − x₁)(x − x₂) when Δ ≥ 0.
2×2 system
{ ax + by = e ; cx + dy = f }. Solve by substitution or linear combination.
Determinant method (Cramer)
D = ad − bc. If D ≠ 0: x = (ed−bf)/D, y = (af−ec)/D.
Theorem 1 - Equations and Inequalities
Application of theorem 1.
Theorem 2 - Equations and Inequalities
Application of theorem 2.
Theorem 3 - Equations and Inequalities
Application of theorem 3.
Theorem 4 - Equations and Inequalities
Application of theorem 4.
Theorem 5 - Equations and Inequalities
Application of theorem 5.
Problem-solving method
Identify data, set formulas, solve step by step, then verify sign, unit, and coherence.
Common mistakes
Forgetting domain constraints, mixing equality/equivalence, and losing solutions during transformations.
Canonical quadratic form
For ax²+bx+c, use Δ=b²-4ac to determine the number of roots.
Linear systems
Usual methods: substitution, elimination, and graphical interpretation in the plane.