AcademyReal Numbers
Academy
Algebraic Manipulation
Level 1 - Math I (Physics) topic page in Real Numbers.
Expanding Brackets
When expanding brackets, each term inside the bracket is multiplied by the term(s) outside.
Single Term Outside
Single Term Expansion
\[a(b + c) = ab + ac\]
Example: \(3(x + 4) = 3x + 12\)
Two Terms Outside (FOIL)
For \((a + b)(c + d)\), multiply:
- First: \(a \times c\)
- Outer: \(a \times d\)
- Inner: \(b \times c\)
- Last: \(b \times d\)
FOIL
\[(a + b)(c + d) = ac + ad + bc + bd\]
Special Products
Difference of Squares
\[(a + b)(a - b) = a^2 - b^2\]
Perfect Square (Sum)
\[(a + b)^2 = a^2 + 2ab + b^2\]
Perfect Square (Difference)
\[(a - b)^2 = a^2 - 2ab + b^2\]
Cube Sum
\[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]
Factoring
Factoring is the reverse of expanding—expressing an expression as a product of factors.
Common Factor
Factor out the greatest common factor (GCF):
Common Factor
\[ax + ay = a(x + y)\]
Difference of Squares
Factor Difference of Squares
\[a^2 - b^2 = (a + b)(a - b)\]
Quadratic Trinomials
For \(ax^2 + bx + c\), find two numbers that multiply to \(ac\) and add to \(b\):
Quadratic Factoring
\[x^2 + 5x + 6 = (x + 2)(x + 3)\]
Completing the Square
Convert a quadratic \(ax^2 + bx + c\) to vertex form:
Completing Square
\[x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2\]
General form:
Vertex Form
\[ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4ac}{4a}\]
This form reveals the vertex of the parabola at \(\left(-\frac{b}{2a}, -\frac{b^2 - 4ac}{4a}\right)\).
Strategy for Algebraic Manipulation
- Identify the goal — What form do you need?
- Simplify first — Combine like terms, factor out GCF
- Choose the technique — Expand, factor, or complete the square
- Verify — Check by expanding your result