Algebra basics and manipulation
Working with algebra
- In algebra, letters stand for numbers; a variable can change value.
- To substitute is to put a number in place of a letter.
- Example: $3x^2 - 2y$ at $x=4,\ y=5$ gives $3(16) - 10 = 38$.
Practice
Find the value of 3x² − 2y when x = 4 and y = 5.
3 × 4² − 2 × 5 = 3 × 16 − 10 = 48 − 10 = 38.
Simplify, expand, factorise
- Like terms can be added: $2a^2 + 5a^2 = 7a^2$.
- Expand brackets: $(2x+1)(x-4) = 2x^2 - 7x - 4$.
- Factorise (the reverse): take out a common factor, e.g. $9x^2 + 15xy = 3x(3x + 5y)$.
- (Extended) difference of two squares: $9x^2 - 16 = (3x+4)(3x-4)$; completing the square: $x^2 + 6x + 1 = (x+3)^2 - 8$.
Practice
Expand (2x + 1)(x − 4) = 2x² + bx + c. What is the constant term c?
(2x+1)(x−4) = 2x² − 8x + x − 4 = 2x² − 7x − 4, so c = −4.
Practice
Factorise the difference of two squares 9x² − 16.
a² − b² = (a+b)(a−b), with a = 3x, b = 4.
You've got it
Key idea
- substitute numbers for letters; collect like terms
- expand = multiply out; factorise = take out common factors / brackets
- (Extended) $a^2 - b^2 = (a+b)(a-b)$; complete the square to $(x+p)^2 + q$