Year 10 Mathematics Revision: Algebra & Logarithms

This cheat sheet covers essential Year 10 mathematics topics including simplifying algebraic expressions, expanding and factorizing polynomials, and evaluating and solving logarithmic equations.

Core Principles

• Algebraic Simplification: Combine like terms, use exponent rules, and distribute to simplify expressions.
• Expansion: Apply the distributive property (FOIL) or binomial expansion formulas to multiply polynomials.
• Factorization: Find common factors, use difference of squares, or factor quadratic expressions.
• Logarithm Properties: Understand base, argument, and the rules for addition, subtraction, and multiplication.
• Solving Logarithmic Equations: Convert logarithmic form to exponential form or use logarithm properties to isolate the variable.
• Completing the Square: Manipulate quadratic expressions into the form $(x+a)^2 + b$ for factorization or graphing.

Formulas

• $(a+b)^2 = a^2 + 2ab + b^2$
• $(a-b)^2 = a^2 - 2ab + b^2$
• $a^2 - b^2 = (a-b)(a+b)$
• $\\log_b(x) + \\log_b(y) = \\log_b(xy)$
• $\\log_b(x) - \\log_b(y) = \\log_b(x/y)$
• $\\log_b(x^n) = n \\log_b(x)$
• $\\log_b(x) = y \\iff b^y = x$

Quiz

• Simplify: $7a^3 - (2a^2)^2 + 5$: $7a^3 - 4a^4 + 5$
• Evaluate: $\\log_2(8) + \\log_2(4)$: 5
• Factorize: $3x^2 - 12x$: $3x(x-4)$