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Key Concepts
Algebra – Linear and Quadratic Equations – Complete Study Notes
Algebra is the branch of mathematics that uses symbols, letters, and numbers to express relationships and solve problems. It is the gateway to higher mathematics and has applications in physics, economics, computer science, and daily life. The word "algebra" comes from the Arabic word al-jabr, meaning "reunion of broken parts."
1. Linear Equations
A linear equation is one where the highest power of the variable is 1. Its graph is a straight line.
- Standard form: ax + b = 0, where a ≠ 0.
- Solution: x = –b/a
- Example: 2x + 6 = 0 → x = –3
- Two-variable linear equation: ax + by = c. These represent lines in a plane. Two such equations form a system — solved by substitution, elimination, or graphically.
2. Quadratic Equations
A quadratic equation has the highest power of 2: ax² + bx + c = 0, where a ≠ 0.
- Quadratic Formula: x = (–b ± √(b²–4ac)) / 2a
- Discriminant (D) = b² – 4ac:
- D > 0: Two distinct real roots
- D = 0: Two equal (repeated) real roots
- D < 0: No real roots (two complex roots)
- Sum of roots: α + β = –b/a
- Product of roots: αβ = c/a
Methods to solve quadratic equations:
- Factorization: Split the middle term and factor.
- Completing the Square: Transform into (x + h)² = k form.
- Quadratic Formula: Always works.
3. Polynomials
A polynomial is an expression with multiple terms involving variables and coefficients: p(x) = aₙxⁿ + ... + a₁x + a₀.
- Degree = highest power of variable.
- Factor Theorem: (x – a) is a factor of p(x) if p(a) = 0.
- Remainder Theorem: When p(x) is divided by (x – a), the remainder = p(a).
4. Arithmetic Progressions (AP)
- Sequence: a, a+d, a+2d, ... where a = first term, d = common difference.
- nᵗʰ term: aₙ = a + (n–1)d
- Sum of n terms: Sₙ = n/2 × [2a + (n–1)d] = n/2 × (first + last term)
5. Functions and Relations
A function is a relation where every element of the domain has exactly one image in the range. Key function types include linear (f(x) = mx + c), quadratic (f(x) = ax²), and exponential (f(x) = aˣ).
- Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.
- Domain: Set of all valid input values.
- Range: Set of all output values.
6. Logarithms
- If aˣ = b, then x = log_a(b)
- log(mn) = log m + log n
- log(m/n) = log m – log n
- log(mⁿ) = n log m
- log_a(a) = 1 and log_a(1) = 0
Key Exam Tips
- Always check the discriminant before stating nature of roots.
- Sum of roots = –b/a; Product = c/a — useful for verify answers.
- In APs, identify first term (a) and common difference (d) first.
- log is only defined for positive numbers and positive base ≠ 1.